🧠:[<think>Okay, let me try to work through this problem step by step. It's about a hyperbola with the equation ( frac{x^2}{a^2} - frac{y^2}{b^2} = 1 ), where ( a > b > 0 ). The problem involves finding the difference in the eccentricity ( f(theta) ) when ( theta ) is ( frac{2pi}{3} ) and ( frac{pi}{3} ). The specific question is to find ( fleft(frac{2pi}{3}right) - fleft(frac{pi}{3}right) ). First, let me recall some basic properties of hyperbolas. The standard form is ( frac{x^2}{a^2} - frac{y^2}{b^2} = 1 ), with the transverse axis along the x-axis. The foci are located at ( (pm c, 0) ), where ( c^2 = a^2 + b^2 ). The eccentricity ( e ) is given by ( e = frac{c}{a} ). Since ( c > a ), the eccentricity of a hyperbola is always greater than 1.Given that ( F ) is the right focus, its coordinates should be ( (c, 0) ). Now, the problem states that we need to draw a perpendicular line to the x-axis through ( F ). A perpendicular line to the x-axis is a vertical line. The equation of this vertical line passing through ( F ) is ( x = c ). This line intersects the hyperbola at points ( M ) and ( N ). So, to find the coordinates of ( M ) and ( N ), we can substitute ( x = c ) into the hyperbola equation and solve for ( y ).Let me do that. Substitute ( x = c ) into ( frac{x^2}{a^2} - frac{y^2}{b^2} = 1 ):[frac{c^2}{a^2} - frac{y^2}{b^2} = 1]Since ( c^2 = a^2 + b^2 ), substitute that in:[frac{a^2 + b^2}{a^2} - frac{y^2}{b^2} = 1 1 + frac{b^2}{a^2} - frac{y^2}{b^2} = 1 frac{b^2}{a^2} - frac{y^2}{b^2} = 0 ]Multiply both sides by ( a^2 b^2 ) to eliminate denominators:[b^4 - a^2 y^2 = 0 a^2 y^2 = b^4 y^2 = frac{b^4}{a^2} y = pm frac{b^2}{a}]Therefore, the points ( M ) and ( N ) are ( (c, frac{b^2}{a}) ) and ( (c, -frac{b^2}{a}) ). Next, the problem mentions point ( A ), the left vertex. The left vertex of the hyperbola is at ( (-a, 0) ). We need to find the angle ( theta = angle MAN ). Points ( M ) and ( N ) are known, ( A ) is known, so we need to figure out the angle between the two lines ( AM ) and ( AN ).To find ( theta ), which is the angle between vectors ( vec{AM} ) and ( vec{AN} ), we can use the dot product formula. The angle between two vectors ( vec{u} ) and ( vec{v} ) is given by:[costheta = frac{vec{u} cdot vec{v}}{|vec{u}| |vec{v}|}]First, let me find the coordinates of vectors ( vec{AM} ) and ( vec{AN} ). Point ( A ) is ( (-a, 0) ). Point ( M ) is ( (c, frac{b^2}{a}) ), so vector ( vec{AM} ) is ( (c - (-a), frac{b^2}{a} - 0) = (c + a, frac{b^2}{a}) ).Similarly, point ( N ) is ( (c, -frac{b^2}{a}) ), so vector ( vec{AN} ) is ( (c - (-a), -frac{b^2}{a} - 0) = (c + a, -frac{b^2}{a}) ).Now, compute the dot product ( vec{AM} cdot vec{AN} ):[(c + a)(c + a) + left( frac{b^2}{a} right)left( -frac{b^2}{a} right) = (c + a)^2 - left( frac{b^4}{a^2} right)]Compute the magnitudes of ( vec{AM} ) and ( vec{AN} ):Since both vectors ( vec{AM} ) and ( vec{AN} ) have the same magnitude:[|vec{AM}| = |vec{AN}| = sqrt{(c + a)^2 + left( frac{b^2}{a} right)^2}]Therefore, the cosine of the angle ( theta ) is:[costheta = frac{(c + a)^2 - frac{b^4}{a^2}}{(c + a)^2 + frac{b^4}{a^2}}]Let me simplify this expression. Let's denote ( S = (c + a)^2 ) and ( T = frac{b^4}{a^2} ), then ( costheta = frac{S - T}{S + T} ).But let's compute this explicitly. First, expand ( (c + a)^2 ):[(c + a)^2 = c^2 + 2ac + a^2]But since ( c^2 = a^2 + b^2 ), substitute:[(c + a)^2 = a^2 + b^2 + 2ac + a^2 = 2a^2 + b^2 + 2ac]Therefore, the numerator becomes:[2a^2 + b^2 + 2ac - frac{b^4}{a^2}]And the denominator is:[2a^2 + b^2 + 2ac + frac{b^4}{a^2}]This looks complicated. Let's see if we can express everything in terms of ( e ), the eccentricity, since the problem mentions that the eccentricity is ( f(theta) ), so we need to relate ( e ) to ( theta ).Recall that ( e = frac{c}{a} ), so ( c = a e ). Also, since ( c^2 = a^2 + b^2 ), we can write:[b^2 = c^2 - a^2 = a^2 e^2 - a^2 = a^2 (e^2 - 1)]Therefore, ( frac{b^2}{a} = a (e^2 - 1) ). Wait, let's check that. If ( b^2 = a^2 (e^2 - 1) ), then ( frac{b^2}{a} = a (e^2 - 1) ), right. So, perhaps substituting ( b^2 = a^2 (e^2 - 1) ) into our expressions for numerator and denominator will help.First, let's write ( c = a e ). Then ( c + a = a e + a = a (e + 1) ). Therefore, ( (c + a)^2 = a^2 (e + 1)^2 ).Also, ( frac{b^4}{a^2} = frac{[a^2 (e^2 - 1)]^2}{a^2} = frac{a^4 (e^2 - 1)^2}{a^2} = a^2 (e^2 - 1)^2 ).So substituting into the numerator and denominator:Numerator: ( (c + a)^2 - frac{b^4}{a^2} = a^2 (e + 1)^2 - a^2 (e^2 - 1)^2 = a^2 [ (e + 1)^2 - (e^2 - 1)^2 ] )Denominator: ( (c + a)^2 + frac{b^4}{a^2} = a^2 (e + 1)^2 + a^2 (e^2 - 1)^2 = a^2 [ (e + 1)^2 + (e^2 - 1)^2 ] )Therefore, the ratio is:[frac{ (e + 1)^2 - (e^2 - 1)^2 }{ (e + 1)^2 + (e^2 - 1)^2 }]Let me compute the numerator and denominator separately.First, expand the numerator:( (e + 1)^2 - (e^2 - 1)^2 )Expand both squares:( (e^2 + 2e + 1) - (e^4 - 2e^2 + 1) )Simplify:( e^2 + 2e + 1 - e^4 + 2e^2 - 1 )Combine like terms:- ( -e^4 )- ( e^2 + 2e^2 = 3e^2 )- ( 2e )- ( 1 - 1 = 0 )So numerator is ( -e^4 + 3e^2 + 2e )Similarly, denominator:( (e + 1)^2 + (e^2 - 1)^2 )Expand both squares:( e^2 + 2e + 1 + e^4 - 2e^2 + 1 )Simplify:( e^4 - e^2 + 2e + 2 )So denominator is ( e^4 - e^2 + 2e + 2 )Therefore, we have:[costheta = frac{ -e^4 + 3e^2 + 2e }{ e^4 - e^2 + 2e + 2 }]Hmm, this is a ratio of two polynomials in ( e ). Let me check my algebra to ensure I didn't make a mistake.Starting with the numerator:( (e + 1)^2 = e^2 + 2e + 1 )( (e^2 - 1)^2 = e^4 - 2e^2 + 1 )Subtracting the second from the first:( e^2 + 2e + 1 - e^4 + 2e^2 - 1 = ( -e^4 ) + (e^2 + 2e^2) + (2e) + (1 - 1) = -e^4 + 3e^2 + 2e ). That seems correct.Denominator:( (e + 1)^2 + (e^2 - 1)^2 = e^2 + 2e + 1 + e^4 - 2e^2 + 1 = e^4 + (e^2 - 2e^2) + 2e + (1 + 1) = e^4 - e^2 + 2e + 2 ). Also correct.So, ( costheta = frac{ -e^4 + 3e^2 + 2e }{ e^4 - e^2 + 2e + 2 } )This equation relates ( costheta ) to the eccentricity ( e ). Since we need ( e ) as a function of ( theta ), denoted ( f(theta) ), we need to solve for ( e ) in terms of ( theta ). However, solving this equation for ( e ) in terms of ( theta ) seems non-trivial because it's a quartic equation. Perhaps there's a smarter way to approach this.Alternatively, maybe we can factor the numerator and denominator.Looking at the numerator: ( -e^4 + 3e^2 + 2e ). Let me factor out a negative sign: ( - (e^4 - 3e^2 - 2e) ). Hmm, not sure if that factors nicely.Denominator: ( e^4 - e^2 + 2e + 2 ). Also not obviously factorable.Alternatively, perhaps we can write the numerator and denominator in terms of ( e^2 ).But even so, it's still complex. Let me think if there's another approach. Maybe using parametric coordinates or geometric properties of the hyperbola.Alternatively, since we need to evaluate ( f(theta) ) at ( theta = frac{2pi}{3} ) and ( theta = frac{pi}{3} ), and then subtract, perhaps we can compute ( e ) for both angles and find the difference. So maybe plug ( theta = frac{2pi}{3} ) into the equation and solve for ( e ), then do the same for ( theta = frac{pi}{3} ), then subtract the two.But solving the quartic equation might be complicated. Let's see:First, for ( theta = frac{2pi}{3} ), ( costheta = cosleft(frac{2pi}{3}right) = -frac{1}{2} ). Similarly, for ( theta = frac{pi}{3} ), ( costheta = cosleft(frac{pi}{3}right) = frac{1}{2} ).So, set up the equation:For ( theta = frac{2pi}{3} ):[-frac{1}{2} = frac{ -e^4 + 3e^2 + 2e }{ e^4 - e^2 + 2e + 2 }]Multiply both sides by denominator:[-frac{1}{2} (e^4 - e^2 + 2e + 2) = -e^4 + 3e^2 + 2e]Multiply out the left side:[-frac{1}{2}e^4 + frac{1}{2}e^2 - e - 1 = -e^4 + 3e^2 + 2e]Multiply both sides by 2 to eliminate fractions:[- e^4 + e^2 - 2e - 2 = -2e^4 + 6e^2 + 4e]Bring all terms to one side:[- e^4 + e^2 - 2e - 2 + 2e^4 - 6e^2 - 4e = 0 ( -e^4 + 2e^4 ) + ( e^2 - 6e^2 ) + ( -2e - 4e ) + ( -2 ) = 0 e^4 -5e^2 -6e -2 = 0]So, the equation becomes:[e^4 -5e^2 -6e -2 = 0]This is a quartic equation. Hmm, solving quartic equations is not straightforward. Maybe there is a rational root. Let's try possible rational roots using Rational Root Theorem. Possible roots are ±1, ±2.Test e=1: 1 -5 -6 -2 = -12 ≠ 0e= -1: 1 -5 +6 -2 = 0 → 0! So e=-1 is a root.But eccentricity is always positive, greater than 1, so e=-1 is invalid, but maybe factor out (e + 1):Divide polynomial by (e + 1):Using synthetic division:Coefficients: 1 (e^4), 0 (e^3), -5 (e^2), -6 (e), -2 (constant)Root: -1Bring down 1. Multiply by -1: -1. Add to next term: 0 + (-1) = -1.Multiply by -1: 1. Add to next term: -5 +1 = -4.Multiply by -1: 4. Add to next term: -6 +4 = -2.Multiply by -1: 2. Add to last term: -2 +2 = 0.So the quotient polynomial is e^3 - e^2 -4e -2.So, the equation factors as (e +1)(e^3 - e^2 -4e -2)=0.Now, check if cubic has any rational roots. Possible roots: ±1, ±2.Test e=2: 8 -4 -8 -2 = -6 ≠0e= -1: -1 -1 +4 -2=0. So e=-1 is again a root.But since e>0, discard. So factor cubic as (e +1)(e^2 -2e -2).Wait, divide e^3 -e^2 -4e -2 by (e +1):Using synthetic division:Coefficients: 1, -1, -4, -2Root: -1Bring down 1. Multiply by -1: -1. Add to next term: -1 + (-1) = -2.Multiply by -1: 2. Add to next term: -4 +2 = -2.Multiply by -1: 2. Add to last term: -2 +2=0.Thus, cubic factors as (e +1)(e^2 -2e -2). Therefore, original equation factors as (e +1)^2 (e^2 -2e -2)=0.Therefore, roots are e=-1 (double root), and roots of e^2 -2e -2=0. Solving this quadratic:e = [2 ± sqrt(4 + 8)] / 2 = [2 ± sqrt(12)] / 2 = [2 ± 2*sqrt(3)] / 2 = 1 ± sqrt(3). Since eccentricity e>1, so e=1 + sqrt(3) ≈ 2.732. The other root is 1 - sqrt(3) ≈ -0.732, which is negative, so invalid. Therefore, for θ=2π/3, the eccentricity is e=1 + sqrt(3).Similarly, now for θ=π/3, which has cosθ=1/2.Set up the equation:1/2 = ( -e^4 +3e^2 +2e ) / ( e^4 -e^2 +2e +2 )Multiply both sides by denominator:(1/2)(e^4 -e^2 +2e +2) = -e^4 +3e^2 +2eMultiply both sides by 2:e^4 -e^2 +2e +2 = -2e^4 +6e^2 +4eBring all terms to left side:e^4 -e^2 +2e +2 +2e^4 -6e^2 -4e =0Combine like terms:(1+2)e^4 + (-1 -6)e^2 + (2 -4)e +2 =03e^4 -7e^2 -2e +2=0Again, a quartic equation. Let's attempt to find rational roots. Possible roots: ±1, ±2.Test e=1: 3 -7 -2 +2= -4≠0e=2: 3*16 -7*4 -4 +2=48 -28 -4 +2=18≠0e= -1: 3 +7 +2 +2=14≠0e= -2: 3*16 -7*4 +4 +2=48 -28 +4 +2=26≠0No rational roots. Maybe factor by grouping or other methods. Alternatively, let's try to factor this quartic.Alternatively, try substituting t = e^2. Then equation becomes:3t^2 -7t -2e +2=0. Hmm, but still has a term with e. Maybe not helpful.Alternatively, perhaps assume that this quartic can be factored into quadratics. Let's suppose:3e^4 -7e^2 -2e +2 = (ae^2 + be + c)(de^2 + fe + g)Multiply out:ade^4 + (af + bd)e^3 + (ag + bf + cd)e^2 + (bg + cf)e + cgComparing coefficients:ad = 3af + bd =0 (since there is no e^3 term)ag + bf + cd = -7bg + cf = -2cg = 2We need integer coefficients. Let's try a=3, d=1 (since 3*1=3). Then:ad=3, so a=3, d=1.Now, af + bd =0 → 3f + b*1 =0 → 3f + b =0 ⇒ b= -3fNext, cg=2. Possible integer pairs: (c,g)=(1,2),(2,1),(-1,-2),(-2,-1)Try c=2, g=1:Then, bg + cf = -2 ⇒ b*1 + 2f = -2. Since b= -3f, substitute:-3f +2f = -2 ⇒ -f = -2 ⇒ f=2Thus, b= -3*2= -6Now check ag + bf + cd = -7:a=3, g=1, b= -6, f=2, c=2, d=1ag + bf + cd =3*1 + (-6)*2 +2*1=3 -12 +2= -7. Perfect.So, the quartic factors as (3e^2 -6e +2)(e^2 +2e +1). Let's verify:(3e^2 -6e +2)(e^2 +2e +1) = 3e^4 +6e^3 +3e^2 -6e^3 -12e^2 -6e +2e^2 +4e +2Combine like terms:3e^4 + (6e^3 -6e^3) + (3e^2 -12e^2 +2e^2) + (-6e +4e) +2 = 3e^4 -7e^2 -2e +2. Correct.Therefore, the quartic factors as (3e^2 -6e +2)(e^2 +2e +1)=0. Thus, set each factor to zero:e^2 +2e +1=0 ⇒ (e+1)^2=0 ⇒ e=-1 (double root). Disregard since e>1.3e^2 -6e +2=0 ⇒ e=(6±sqrt(36 -24))/6=(6±sqrt(12))/6=(6±2sqrt(3))/6=1±(sqrt(3)/3). Compute numerical values:sqrt(3)/3≈0.577, so e≈1 + 0.577≈1.577 or e≈1 -0.577≈0.423. Since eccentricity must be >1, so e=1 + sqrt(3)/3≈1.577. Therefore, for θ=π/3, the eccentricity is e=1 + sqrt(3)/3.So now, we have:f(2π/3)=1 + sqrt(3)f(π/3)=1 + sqrt(3)/3Therefore, the difference is:[1 + sqrt(3)] - [1 + sqrt(3)/3] = sqrt(3) - sqrt(3)/3 = (2/3)sqrt(3)Rationalizing, that's (2 sqrt(3))/3.Hence, the answer should be ( boxed{dfrac{2sqrt{3}}{3}} ).But let me verify again to make sure I didn't make a mistake in solving the equations.For θ=2π/3, we had:Quartic equation e^4 -5e^2 -6e -2=0, which factored into (e +1)^2 (e^2 -2e -2)=0, leading to e=1 + sqrt(3)≈2.732.For θ=π/3, equation became 3e^4 -7e^2 -2e +2=0, which factored into (3e^2 -6e +2)(e +1)^2=0, leading to e=1 + sqrt(3)/3≈1.577.Thus, the difference is 1 + sqrt(3) -1 - sqrt(3)/3= sqrt(3) - sqrt(3)/3= (2 sqrt(3))/3.Yes, that seems correct.But just to make sure, let's check with another approach. Maybe compute the angle θ using coordinates and see if with e=1 + sqrt(3), the angle is indeed 2π/3.Let me take e=1 + sqrt(3). Then compute coordinates:First, a is arbitrary, since the angle does not depend on a (because angles are invariant under scaling). Let me set a=1 for simplicity.Then, c = a e = e =1 + sqrt(3). b^2 =a^2 (e^2 -1)=1*( (1 + sqrt(3))^2 -1 )=1*(1 +2sqrt(3) +3 -1)=1*(3 +2sqrt(3))=3 +2sqrt(3)Therefore, points M and N are (c, b^2/a)=(1 + sqrt(3), 3 + 2sqrt(3)) and (1 + sqrt(3), -3 -2sqrt(3)). Wait, wait, no.Wait, points M and N are (c, b^2/a). Wait, from earlier, when x=c, y=±b^2/a. So since a=1, y=±b^2. But b^2=3 +2sqrt(3). So points M=(1 + sqrt(3), 3 +2sqrt(3)), N=(1 + sqrt(3), -3 -2sqrt(3)).Point A is (-a,0)=(-1,0). So vectors AM and AN are:AM = (1 + sqrt(3) - (-1), 3 +2sqrt(3)-0)=(2 + sqrt(3), 3 +2sqrt(3))AN = (2 + sqrt(3), -3 -2sqrt(3))Compute the angle between these vectors.The dot product is (2 + sqrt(3))^2 + (3 +2sqrt(3))*(-3 -2sqrt(3)).First, compute (2 + sqrt(3))^2 =4 +4sqrt(3)+3=7 +4sqrt(3)Then, compute (3 +2sqrt(3))*(-3 -2sqrt(3)) = -9 -6sqrt(3) -6sqrt(3) -4*3 = -9 -12sqrt(3) -12= -21 -12sqrt(3)So total dot product:7 +4sqrt(3) -21 -12sqrt(3)= -14 -8sqrt(3)Magnitude of AM: sqrt( (2 + sqrt(3))^2 + (3 +2sqrt(3))^2 )First compute (2 + sqrt(3))^2=7 +4sqrt(3)(3 +2sqrt(3))^2=9 +12sqrt(3)+12=21 +12sqrt(3)Sum:7 +4sqrt(3) +21 +12sqrt(3)=28 +16sqrt(3)Therefore, |AM|=sqrt(28 +16sqrt(3)). Similarly, |AN|=same value.So, cosθ=( -14 -8sqrt(3) )/(28 +16sqrt(3)).Factor numerator and denominator:Numerator: -2*(7 +4sqrt(3))Denominator: 4*(7 +4sqrt(3))Thus, cosθ= (-2/4)= -1/2. Therefore, θ=2π/3. Correct. So, e=1 + sqrt(3) gives θ=2π/3.Similarly, check for θ=π/3, e=1 + sqrt(3)/3≈1.577.Let me set a=1, e=1 + sqrt(3)/3≈1.577, then c=1 + sqrt(3)/3.Compute b^2= a^2(e^2 -1)= ( (1 + sqrt(3)/3)^2 -1 )=1 + (2 sqrt(3))/3 + (3)/9 -1= (2 sqrt(3))/3 + 1/3 -0= (2 sqrt(3) +1)/3.Wait, let's compute step by step:e=1 + sqrt(3)/3. Then e^2=1 + (2 sqrt(3))/3 + (3)/9=1 + (2 sqrt(3))/3 + 1/3= (3 +2 sqrt(3) +1)/3=(4 +2 sqrt(3))/3.Then e^2 -1=(4 +2 sqrt(3))/3 -1=(4 +2 sqrt(3) -3)/3=(1 +2 sqrt(3))/3.Therefore, b^2=(1 +2 sqrt(3))/3.Then, points M and N are (c, b^2/a)= (1 + sqrt(3)/3, (1 +2 sqrt(3))/3 )Point A is (-1,0). So vector AM is (1 + sqrt(3)/3 +1, (1 +2 sqrt(3))/3 -0)=(2 + sqrt(3)/3, (1 +2 sqrt(3))/3 )Vector AN is (2 + sqrt(3)/3, -(1 +2 sqrt(3))/3 )Compute the dot product:(2 + sqrt(3)/3)(2 + sqrt(3)/3) + [(1 +2 sqrt(3))/3][ - (1 +2 sqrt(3))/3 ]First term: (2 + sqrt(3)/3)^2=4 + (4 sqrt(3))/3 + (3)/9=4 + (4 sqrt(3))/3 + 1/3= (12 +4 sqrt(3) +1)/3=(13 +4 sqrt(3))/3Second term: -[(1 +2 sqrt(3))/3]^2= -[1 +4 sqrt(3) +12]/9= -[13 +4 sqrt(3)]/9So total dot product:(13 +4 sqrt(3))/3 - (13 +4 sqrt(3))/9 = [ (39 +12 sqrt(3)) -13 -4 sqrt(3) ]/9= (26 +8 sqrt(3))/9Compute magnitude squared of vectors AM:(2 + sqrt(3)/3)^2 + [(1 +2 sqrt(3))/3]^2= (13 +4 sqrt(3))/3 + (1 +4 sqrt(3) +12)/9= (13 +4 sqrt(3))/3 + (13 +4 sqrt(3))/9= (39 +12 sqrt(3) +13 +4 sqrt(3))/9= (52 +16 sqrt(3))/9Therefore, |AM|=sqrt( (52 +16 sqrt(3))/9 )=sqrt(52 +16 sqrt(3))/3Thus, cosθ= (26 +8 sqrt(3))/9 divided by (52 +16 sqrt(3))/9= (26 +8 sqrt(3))/(52 +16 sqrt(3))= factor numerator and denominator:Numerator: 2*(13 +4 sqrt(3))Denominator:4*(13 +4 sqrt(3))Thus, cosθ=2/4=1/2, so θ=π/3. Correct.Therefore, our previous results hold. Therefore, the difference is indeed (2 sqrt(3))/3.Thus, the answer is boxed{dfrac{2sqrt{3}}{3}}.Final Answerboxed{dfrac{2sqrt{3}}{3}}</think>Given the hyperbola ( C: frac{x^2}{a^2} - frac{y^2}{b^2} = 1 ) with ( a > b > 0 ), we need to find the difference ( fleft(frac{2pi}{3}right) - fleft(frac{pi}{3}right) ) where ( f(theta) ) is the eccentricity of the hyperbola.1. Coordinates of Points: - The right focus ( F ) is at ( (c, 0) ) where ( c = sqrt{a^2 + b^2} ). - The perpendicular line through ( F ) intersects the hyperbola at points ( M ) and ( N ) with coordinates ( (c, pm frac{b^2}{a}) ). - The left vertex ( A ) is at ( (-a, 0) ).2. Vectors and Dot Product: - Vectors ( vec{AM} ) and ( vec{AN} ) are ( (c + a, frac{b^2}{a}) ) and ( (c + a, -frac{b^2}{a}) ). - The dot product ( vec{AM} cdot vec{AN} ) and magnitudes ( |vec{AM}| ) and ( |vec{AN}| ) are calculated.3. Cosine of the Angle: - Using the dot product formula, ( costheta ) is expressed in terms of ( e ) (eccentricity).4. Solving for Eccentricity: - For ( theta = frac{2pi}{3} ), ( costheta = -frac{1}{2} ), leading to solving the quartic equation ( e^4 - 5e^2 - 6e - 2 = 0 ). The valid solution is ( e = 1 + sqrt{3} ). - For ( theta = frac{pi}{3} ), ( costheta = frac{1}{2} ), leading to solving the quartic equation ( 3e^4 - 7e^2 - 2e + 2 = 0 ). The valid solution is ( e = 1 + frac{sqrt{3}}{3} ).5. Difference in Eccentricities: - The difference ( fleft(frac{2pi}{3}right) - fleft(frac{pi}{3}right) ) is calculated as ( (1 + sqrt{3}) - (1 + frac{sqrt{3}}{3}) = frac{2sqrt{3}}{3} ).Thus, the final answer is (boxed{dfrac{2sqrt{3}}{3}}).,<think>Okay, so I need to solve this problem about a hyperbola. Let me start by understanding what's given and what's being asked.The hyperbola is given by the equation ( frac{x^{2}}{a^{2}} - frac{y^{2}}{b^{2}} = 1 ) where ( a > b > 0 ). The right focus is labeled as ( F ). Then, they draw a perpendicular line to the x-axis through ( F ), which I assume is a vertical line since it's perpendicular to the x-axis. The points where this vertical line intersects the hyperbola are ( M ) and ( N ). The left vertex is ( A ), and the angle ( angle MAN = theta ). The problem states that the eccentricity of the hyperbola is a function of ( theta ), called ( f(theta) ). We need to find ( f(frac{2pi}{3}) - f(frac{pi}{3}) ).First, let's recall some basic properties of hyperbolas. For a hyperbola ( frac{x^{2}}{a^{2}} - frac{y^{2}}{b^{2}} = 1 ), the foci are located at ( (pm c, 0) ), where ( c^2 = a^2 + b^2 ). The eccentricity ( e ) is given by ( e = frac{c}{a} ). Since ( c > a ), the eccentricity ( e > 1 ).Given that ( F ) is the right focus, its coordinates are ( (c, 0) ). The vertical line through ( F ) is ( x = c ). To find points ( M ) and ( N ), we need to find where this line intersects the hyperbola. Let's substitute ( x = c ) into the hyperbola equation:( frac{c^2}{a^2} - frac{y^2}{b^2} = 1 )Solving for ( y^2 ):( frac{y^2}{b^2} = frac{c^2}{a^2} - 1 )But since ( c^2 = a^2 + b^2 ), substitute that in:( frac{y^2}{b^2} = frac{a^2 + b^2}{a^2} - 1 = frac{a^2 + b^2 - a^2}{a^2} = frac{b^2}{a^2} )Multiply both sides by ( b^2 ):( y^2 = frac{b^4}{a^2} )Therefore, ( y = pm frac{b^2}{a} ), so the points ( M ) and ( N ) are ( (c, frac{b^2}{a}) ) and ( (c, -frac{b^2}{a}) ).The left vertex ( A ) is at ( (-a, 0) ). The angle ( angle MAN ) is formed at point ( A ) by the lines ( AM ) and ( AN ). So we need to find the angle between the two lines connecting ( A ) to ( M ) and ( A ) to ( N ).Let me visualize this. Point ( A ) is on the left, ( M ) and ( N ) are vertically above and below the focus ( F ) on the right side of the hyperbola. So lines ( AM ) and ( AN ) are two lines from the left vertex to these upper and lower points on the vertical line through ( F ). The angle between these two lines is ( theta ), and we need to express the eccentricity ( e ) as a function of ( theta ), then compute the difference ( f(2pi/3) - f(pi/3) ).To find ( theta ) in terms of ( e ), or vice versa, I need to find the angle ( theta = angle MAN ) in terms of the coordinates of points ( A ), ( M ), and ( N ).Since angle ( theta ) is formed at point ( A ), we can use vectors to compute the angle. Let me recall that the angle between two vectors ( vec{u} ) and ( vec{v} ) is given by:( cos theta = frac{vec{u} cdot vec{v}}{||vec{u}|| , ||vec{v}||} )So, first, let's find the coordinates of points ( A ), ( M ), and ( N ):- ( A = (-a, 0) )- ( M = (c, frac{b^2}{a}) )- ( N = (c, -frac{b^2}{a}) )Vectors ( vec{AM} ) and ( vec{AN} ) can be found by subtracting coordinates:( vec{AM} = M - A = (c - (-a), frac{b^2}{a} - 0) = (c + a, frac{b^2}{a}) )( vec{AN} = N - A = (c - (-a), -frac{b^2}{a} - 0) = (c + a, -frac{b^2}{a}) )Now, compute the dot product ( vec{AM} cdot vec{AN } ):( (c + a)(c + a) + left( frac{b^2}{a} right)left( -frac{b^2}{a} right) = (c + a)^2 - left( frac{b^4}{a^2} right) )Compute the magnitudes ( ||vec{AM}|| ) and ( ||vec{AN}|| ). Since both vectors have the same magnitude:( ||vec{AM}|| = ||vec{AN}|| = sqrt{(c + a)^2 + left( frac{b^2}{a} right)^2} )Therefore, the cosine of angle ( theta ) is:( cos theta = frac{(c + a)^2 - frac{b^4}{a^2}}{(c + a)^2 + frac{b^4}{a^2}} )Let me simplify this expression. Let's denote ( D = (c + a)^2 ), then:( cos theta = frac{D - frac{b^4}{a^2}}{D + frac{b^4}{a^2}} )Let me express everything in terms of ( a ) and ( e ), since ( e = frac{c}{a} ), so ( c = a e ). Also, ( c^2 = a^2 + b^2 ), so ( b^2 = c^2 - a^2 = a^2(e^2 - 1) ). Therefore, ( b^4 = a^4(e^2 - 1)^2 ).Substitute ( c = a e ) into ( D ):( D = (a e + a)^2 = a^2(e + 1)^2 )Substitute ( D ) and ( b^4 ) into the expression for ( cos theta ):( cos theta = frac{a^2(e + 1)^2 - frac{a^4(e^2 - 1)^2}{a^2}}{a^2(e + 1)^2 + frac{a^4(e^2 - 1)^2}{a^2}} )Simplify numerator and denominator:Numerator: ( a^2(e + 1)^2 - a^2(e^2 - 1)^2 )Denominator: ( a^2(e + 1)^2 + a^2(e^2 - 1)^2 )Factor out ( a^2 ):Numerator: ( a^2 [ (e + 1)^2 - (e^2 - 1)^2 ] )Denominator: ( a^2 [ (e + 1)^2 + (e^2 - 1)^2 ] )Cancel out ( a^2 ):( cos theta = frac{ (e + 1)^2 - (e^2 - 1)^2 }{ (e + 1)^2 + (e^2 - 1)^2 } )Now, let's compute numerator and denominator separately.First, expand ( (e + 1)^2 ):( (e + 1)^2 = e^2 + 2e + 1 )Next, expand ( (e^2 - 1)^2 ):( (e^2 - 1)^2 = e^4 - 2e^2 + 1 )So, numerator:( (e^2 + 2e + 1) - (e^4 - 2e^2 + 1) = e^2 + 2e + 1 - e^4 + 2e^2 -1 )Simplify:( -e^4 + 3e^2 + 2e )Denominator:( (e^2 + 2e + 1) + (e^4 - 2e^2 + 1) = e^4 - e^2 + 2e + 2 )So now, we have:( cos theta = frac{ -e^4 + 3e^2 + 2e }{ e^4 - e^2 + 2e + 2 } )This seems complicated. Maybe there's a better way to simplify this expression. Let me check my steps again to see if I made any mistakes.Wait, maybe expanding the numerator and denominator isn't the best approach. Let me try to factor the expressions instead.Let me note that ( (e + 1)^2 - (e^2 - 1)^2 ). This is of the form ( A^2 - B^2 = (A - B)(A + B) ). Let's apply that.Let ( A = e + 1 ) and ( B = e^2 - 1 ). Then,Numerator: ( (e + 1 - (e^2 - 1))(e + 1 + e^2 - 1) )Simplify each factor:First factor: ( e + 1 - e^2 + 1 = -e^2 + e + 2 )Second factor: ( e + 1 + e^2 - 1 = e^2 + e )So, numerator becomes ( (-e^2 + e + 2)(e^2 + e) )Similarly, denominator is ( (e + 1)^2 + (e^2 - 1)^2 ). Let me check if that can be factored.Hmm, sum of squares doesn't factor easily. Let's compute it:Denominator: ( e^4 - 2e^2 + 1 + e^2 + 2e + 1 = e^4 - e^2 + 2e + 2 ). Wait, but that was our previous result. So perhaps we can factor the denominator as well?Alternatively, maybe we can factor numerator and denominator expressions.Numerator: ( (-e^2 + e + 2)(e^2 + e) )Let me factor each term:First, factor out -1 from the first factor:( - (e^2 - e - 2)(e^2 + e) )Factor ( e^2 - e - 2 ): looking for two numbers that multiply to -2 and add to -1. Those would be -2 and 1. So:( e^2 - e - 2 = (e - 2)(e + 1) )Therefore, numerator:( - (e - 2)(e + 1)(e)(e + 1) = -e(e - 2)(e + 1)^2 )Wait, let me check:Original numerator after factoring:( - (e^2 - e - 2)(e^2 + e) = - ( (e - 2)(e + 1) )( e(e + 1) ) ) = - e(e - 2)(e + 1)^2 )Yes, that's correct.Denominator: ( e^4 - e^2 + 2e + 2 ). Let's try to factor this. Let me attempt to factor:Looking for rational roots using Rational Root Theorem. Possible roots: ±1, ±2. Let me plug in e = 1:( 1 - 1 + 2 + 2 = 4 ≠ 0 )e = -1:( 1 - 1 - 2 + 2 = 0 ). So e = -1 is a root. Let's perform polynomial division.Divide denominator by (e + 1):Using synthetic division:Coefficients: 1 (e^4), 0 (e^3), -1 (e^2), 2 (e), 2 (constant term)Root: -1Bring down 1.Multiply by -1: -1. Add to next term: 0 + (-1) = -1.Multiply by -1: 1. Add to next term: -1 + 1 = 0.Multiply by -1: 0. Add to next term: 2 + 0 = 2.Multiply by -1: -2. Add to last term: 2 + (-2) = 0. So, remainder is 0.Therefore, denominator factors as (e + 1)(e^3 - e^2 + 0e + 2). Let me check if the cubic can be factored further.Check e = -1 again in the cubic:(-1)^3 - (-1)^2 + 0 + 2 = -1 -1 + 2 = 0. So e = -1 is a root again.Divide the cubic by (e + 1):Coefficients: 1 (e^3), -1 (e^2), 0 (e), 2 (constant)Using synthetic division with root -1:Bring down 1.Multiply by -1: -1. Add to next term: -1 + (-1) = -2.Multiply by -1: 2. Add to next term: 0 + 2 = 2.Multiply by -1: -2. Add to last term: 2 + (-2) = 0.Thus, the cubic factors as (e + 1)(e^2 - 2e + 2). So denominator is:(e + 1)^2 (e^2 - 2e + 2)Therefore, denominator factors into (e + 1)^2 (e^2 - 2e + 2).So putting numerator and denominator together:( cos theta = frac{ -e(e - 2)(e + 1)^2 }{ (e + 1)^2 (e^2 - 2e + 2) } )Cancel out (e + 1)^2 terms:( cos theta = frac{ -e(e - 2) }{ e^2 - 2e + 2 } )Simplify numerator:( -e(e - 2) = -e^2 + 2e )Denominator remains ( e^2 - 2e + 2 )So,( cos theta = frac{ -e^2 + 2e }{ e^2 - 2e + 2 } )Let me write this as:( cos theta = frac{ -e^2 + 2e }{ e^2 - 2e + 2 } )Hmm, perhaps we can factor numerator and denominator differently, but let me see if I can manipulate this expression. Let me factor a negative sign in the numerator:( cos theta = frac{ - (e^2 - 2e) }{ e^2 - 2e + 2 } )Let me denote ( u = e^2 - 2e ). Then,( cos theta = frac{ -u }{ u + 2 } )So,( cos theta = frac{ -u }{ u + 2 } )Let me solve for u:Multiply both sides by ( u + 2 ):( cos theta (u + 2) = -u )Expand:( u cos theta + 2 cos theta = -u )Bring terms with u to one side:( u cos theta + u = -2 cos theta )Factor u:( u ( cos theta + 1 ) = -2 cos theta )Therefore,( u = frac{ -2 cos theta }{ cos theta + 1 } )But ( u = e^2 - 2e ), so:( e^2 - 2e = frac{ -2 cos theta }{ cos theta + 1 } )Multiply numerator and denominator by -1:( e^2 - 2e = frac{ 2 cos theta }{ -cos theta - 1 } = frac{ 2 cos theta }{ - ( cos theta + 1 ) } = - frac{ 2 cos theta }{ cos theta + 1 } )So,( e^2 - 2e = - frac{ 2 cos theta }{ cos theta + 1 } )Let me rewrite this equation:( e^2 - 2e + frac{ 2 cos theta }{ cos theta + 1 } = 0 )This is a quadratic in ( e ). Let me see if I can solve for ( e ).But perhaps instead of trying to solve for ( e ), we can find a relationship between ( e ) and ( theta ).Alternatively, let's rearrange the equation:( e^2 - 2e = - frac{ 2 cos theta }{ cos theta + 1 } )Complete the square on the left-hand side:( e^2 - 2e + 1 = 1 - frac{ 2 cos theta }{ cos theta + 1 } )Left-hand side becomes ( (e - 1)^2 )Right-hand side: ( 1 - frac{2 cos theta }{ cos theta + 1 } = frac{ ( cos theta + 1 ) - 2 cos theta }{ cos theta + 1 } = frac{ 1 - cos theta }{ cos theta + 1 } )Therefore,( (e - 1)^2 = frac{ 1 - cos theta }{ cos theta + 1 } )Simplify the right-hand side using the identity ( 1 - cos theta = 2 sin^2 (theta/2) ) and ( 1 + cos theta = 2 cos^2 (theta/2) ):( frac{2 sin^2 (theta/2)}{2 cos^2 (theta/2)} = tan^2 (theta/2) )Thus,( (e - 1)^2 = tan^2 (theta/2) )Take square roots:( e - 1 = pm tan (theta/2) )But since ( e > 1 ) (because it's a hyperbola), we discard the negative solution:( e - 1 = tan (theta/2) )Therefore,( e = 1 + tan left( frac{theta}{2} right) )So, the eccentricity ( e = f(theta) = 1 + tan left( frac{theta}{2} right) )Hence, ( f(theta) = 1 + tan left( frac{theta}{2} right) )Therefore, we need to compute ( fleft( frac{2pi}{3} right) - fleft( frac{pi}{3} right) )Let's compute each term:First, ( fleft( frac{2pi}{3} right) = 1 + tan left( frac{pi}{3} right) )Since ( frac{2pi}{3} / 2 = pi/3 ).Similarly, ( fleft( frac{pi}{3} right) = 1 + tan left( frac{pi}{6} right) )Compute these values:( tan left( frac{pi}{3} right) = sqrt{3} )( tan left( frac{pi}{6} right) = frac{1}{sqrt{3}} )Therefore,( fleft( frac{2pi}{3} right) = 1 + sqrt{3} )( fleft( frac{pi}{3} right) = 1 + frac{1}{sqrt{3}} )Compute the difference:( fleft( frac{2pi}{3} right) - fleft( frac{pi}{3} right) = (1 + sqrt{3}) - left( 1 + frac{1}{sqrt{3}} right) = sqrt{3} - frac{1}{sqrt{3}} )Simplify:( sqrt{3} - frac{1}{sqrt{3}} = frac{3}{sqrt{3}} - frac{1}{sqrt{3}} = frac{2}{sqrt{3}} = frac{2 sqrt{3}}{3} )Therefore, the answer should be ( frac{2 sqrt{3}}{3} ), which can be written as ( boxed{dfrac{2sqrt{3}}{3}} )But let me check my steps again to ensure there's no mistake. Especially the step where I went from the equation involving cos theta to solving for e. Let me verify.We had:( cos theta = frac{ -e^2 + 2e }{ e^2 - 2e + 2 } )Then, through substitution and completing the square, we arrived at ( e = 1 + tan(theta/2) ). Let me confirm this.Alternatively, if we let ( t = tan (theta/2) ), then using the identity ( cos theta = frac{1 - t^2}{1 + t^2} ). Let's see if that aligns with our expression.Given ( e = 1 + t ), then ( t = e - 1 ). Let's substitute into the expression for ( cos theta ):Using the identity:( cos theta = frac{1 - t^2}{1 + t^2} = frac{1 - (e - 1)^2}{1 + (e - 1)^2} )Let me compute numerator and denominator:Numerator: ( 1 - (e^2 - 2e + 1) = 1 - e^2 + 2e -1 = -e^2 + 2e )Denominator: ( 1 + (e^2 - 2e + 1) = 1 + e^2 - 2e +1 = e^2 - 2e + 2 )Therefore, ( cos theta = frac{ -e^2 + 2e }{ e^2 - 2e + 2 } ), which matches the expression we had earlier. So that's correct. Therefore, our derivation is consistent.Thus, the conclusion ( e = 1 + tan (theta/2) ) is correct.Therefore, the difference ( f(2pi/3) - f(pi/3) = left(1 + tan(pi/3)right) - left(1 + tan(pi/6)right) = tan(pi/3) - tan(pi/6) )Compute this:( tan(pi/3) = sqrt{3} ), ( tan(pi/6) = 1/sqrt{3} ), so difference is ( sqrt{3} - 1/sqrt{3} = (3 - 1)/sqrt{3} = 2/sqrt{3} = 2sqrt{3}/3 )Yes, that's correct. So the answer is ( dfrac{2sqrt{3}}{3} ).Final Answerboxed{dfrac{2sqrt{3}}{3}}</think>Given the hyperbola ( C: frac{x^2}{a^2} - frac{y^2}{b^2} = 1 ) with ( a > b > 0 ), we need to find the difference ( fleft(frac{2pi}{3}right) - fleft(frac{pi}{3}right) ) where the eccentricity ( e ) is a function of the angle ( theta = angle MAN ).1. Identify Key Points and Vectors: - The right focus ( F ) is at ( (c, 0) ) where ( c = sqrt{a^2 + b^2} ). - Points ( M ) and ( N ) are the intersections of the vertical line ( x = c ) with the hyperbola, giving coordinates ( (c, frac{b^2}{a}) ) and ( (c, -frac{b^2}{a}) ). - The left vertex ( A ) is at ( (-a, 0) ).2. Vectors and Dot Product: - Vectors ( vec{AM} ) and ( vec{AN} ) are calculated as ( (c + a, frac{b^2}{a}) ) and ( (c + a, -frac{b^2}{a}) ). - The dot product and magnitudes of these vectors lead to the expression for ( cos theta ).3. Simplify Using Eccentricity: - Express ( c ) and ( b ) in terms of ( a ) and ( e ) (eccentricity): ( c = ae ) and ( b^2 = a^2(e^2 - 1) ). - Simplify the expression for ( cos theta ) using these substitutions.4. Trigonometric Identity and Solving for ( e ): - Use trigonometric identities to simplify the expression, leading to ( (e - 1)^2 = tan^2(theta/2) ). - Solving this gives ( e = 1 + tan(theta/2) ).5. Compute the Difference: - For ( theta = frac{2pi}{3} ), ( e = 1 + tan(pi/3) = 1 + sqrt{3} ). - For ( theta = frac{pi}{3} ), ( e = 1 + tan(pi/6) = 1 + frac{1}{sqrt{3}} ). - The difference is ( (1 + sqrt{3}) - (1 + frac{1}{sqrt{3}}) = sqrt{3} - frac{1}{sqrt{3}} = frac{2sqrt{3}}{3} ).The final answer is (boxed{dfrac{2sqrt{3}}{3}}).]

🔑:From the given information, we have M(c, frac{b^{2}}{a}). When theta = frac{2pi}{3}, we have tan{frac{pi}{3}} = frac{frac{b^{2}}{a}}{a + c}, which gives e = sqrt{3} + 1. Therefore, f(frac{2pi}{3}) = sqrt{3} + 1.When theta = frac{pi}{3}, we have tan{frac{pi}{6}} = frac{frac{b^{2}}{a}}{a + c}, which gives e = frac{sqrt{3}}{3} + 1. Therefore, f(frac{pi}{3}) = frac{sqrt{3}}{3} + 1.Thus, f(frac{2pi}{3}) - f(frac{pi}{3}) = frac{2sqrt{3}}{3}.Hence, the answer is boxed{frac{2sqrt{3}}{3}}.This problem requires us to use the definition of eccentricity and the properties of hyperbolas. By calculating the eccentricity at the angles frac{2pi}{3} and frac{pi}{3}, we can find the desired difference. This is a moderately difficult problem.

🧠:[<think>Okay, so I need to calculate (1 - i) squared, right? Hmm, where do I start? Let me recall how to square a binomial. Wait, it's similar to expanding (a - b) squared. The formula is a squared minus 2ab plus b squared. So, applying that here, where a is 1 and b is i. Let me write that out:(1 - i)^2 = (1)^2 - 2*(1)*(i) + (i)^2.Alright, let's compute each term step by step. First term is 1 squared, which is just 1. Second term is 2 times 1 times i, so that's 2i. The third term is i squared. Now, I remember that i is the imaginary unit and by definition, i squared is equal to -1. So substituting that in, the third term becomes -1.Putting it all together:1 - 2i + (-1).Wait, so that simplifies to (1 - 1) - 2i. Then 1 - 1 is 0, so it's 0 - 2i, which is -2i. Is that right? Let me check again.Original expression: (1 - i)^2.First term: 1 squared is 1.Second term: -2*(1)*(i) is -2i.Third term: (i)^2 is -1.So combining them: 1 - 2i - 1. Ah! Wait, I had a sign error when I first wrote it. The third term is -1, so the expression becomes 1 - 2i - 1. Then 1 - 1 cancels out, leaving -2i. Yeah, that's correct. So the final answer should be -2i.But let me verify another way, maybe by using complex multiplication directly. Let's multiply (1 - i) by itself:(1 - i)*(1 - i).First, multiply 1 by 1: 1*1 = 1.Then, multiply 1 by -i: 1*(-i) = -i.Next, multiply -i by 1: -i*1 = -i.Finally, multiply -i by -i: (-i)*(-i) = i squared. Which is again -1.So adding all those up: 1 - i - i + (-1) = 1 - 2i -1 = (1 -1) -2i = 0 -2i = -2i. Yep, same result. So that seems to confirm it.Alternatively, maybe using the polar form? Let's see. Maybe that's overcomplicating, but just to check. The complex number 1 - i is in the fourth quadrant. Its magnitude is sqrt(1^2 + (-1)^2) = sqrt(2). The angle (argument) is -45 degrees or 315 degrees, which is equivalent to -π/4 radians.So in polar form, 1 - i can be written as sqrt(2) * (cos(-π/4) + i sin(-π/4)).When we square this, we square the magnitude and double the angle. So the squared magnitude is (sqrt(2))^2 = 2. The angle becomes 2*(-π/4) = -π/2.So in polar form, squared is 2*(cos(-π/2) + i sin(-π/2)).cos(-π/2) is 0, and sin(-π/2) is -1. So multiplying, we get 2*(0 + i*(-1)) = -2i. Again, same answer. So regardless of the method, it's -2i.Therefore, I think that's the correct answer. Let me just make sure I didn't mix up any signs or operations. When expanding the binomial, the middle term is definitely -2ab, which with a=1 and b=i becomes -2i. The i squared is -1, so combining 1 - 2i -1 cancels the 1 and -1, leaving -2i. Yep, that's consistent.Another way: if I remember that (a - b)^2 = a^2 - 2ab + b^2, then (1 - i)^2 = 1 - 2i + i^2. Then replacing i^2 with -1 gives 1 - 2i -1 = -2i. All methods lead to the same conclusion. So I feel confident now that the answer is indeed -2i.Final AnswerThe result of (1-i)^2 is boxed{-2i}.</think>To calculate ((1 - i)^2), we use the binomial expansion formula ((a - b)^2 = a^2 - 2ab + b^2) where (a = 1) and (b = i):[(1 - i)^2 = (1)^2 - 2 cdot 1 cdot i + (i)^2]Calculating each term step by step:- The first term is (1^2 = 1).- The second term is (-2 cdot 1 cdot i = -2i).- The third term is (i^2 = -1).Combining these terms:[1 - 2i + (-1) = 1 - 2i - 1 = (1 - 1) - 2i = 0 - 2i = -2i]We can verify this by direct multiplication:[(1 - i)(1 - i) = 1 cdot 1 + 1 cdot (-i) + (-i) cdot 1 + (-i) cdot (-i)]Calculating each product:- (1 cdot 1 = 1)- (1 cdot (-i) = -i)- ((-i) cdot 1 = -i)- ((-i) cdot (-i) = i^2 = -1)Adding these products together:[1 - i - i - 1 = (1 - 1) + (-i - i) = 0 - 2i = -2i]Thus, the result of ((1 - i)^2) is (boxed{-2i}).,<think>Okay, so I need to calculate (1 - i) squared, where i is the imaginary unit. Hmm, let me think. I remember that squaring a complex number is similar to squaring a binomial in algebra, but I have to remember that i squared is -1. Let me start by writing it out step by step.First, (1 - i)^2 means (1 - i) multiplied by itself, right? So that's (1 - i)(1 - i). To multiply these two binomials, I should use the distributive property, also known as the FOIL method for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, which refers to the terms you need to multiply.Let's apply that here. The First terms are 1 and 1. Multiplying those gives 1*1 = 1.Next, the Outer terms. That would be the 1 from the first binomial and the -i from the second binomial. So that's 1*(-i) = -i.Then the Inner terms. Those are -i from the first binomial and 1 from the second binomial. Multiplying those gives -i*1 = -i.Finally, the Last terms. Both are -i, so that's (-i)*(-i). Hmm, multiplying those together. Let's see, i times i is i squared, and since both are negative, the negatives will cancel out. So (-i)*(-i) = i^2. But wait, i squared is equal to -1. So that term becomes -1.Now, let's add up all those parts. First was 1, then -i, then another -i, and finally -1. So putting it all together: 1 - i - i -1.Let me combine like terms. The real numbers are 1 and -1. 1 -1 is 0. Then the imaginary terms are -i -i, which is -2i. So adding those together: 0 - 2i, which simplifies to -2i.Wait, let me check that again to make sure I didn't make a mistake. Sometimes with complex numbers, it's easy to slip up with the signs. So expanding (1 - i)^2:First: 1*1 = 1Outer: 1*(-i) = -iInner: (-i)*1 = -iLast: (-i)*(-i) = i^2 = -1Adding them: 1 - i - i -1 = (1 -1) + (-i -i) = 0 - 2i = -2i.Yep, that seems right. So the result is -2i. Let me verify another way, maybe using the formula (a - b)^2 = a^2 - 2ab + b^2. Applying this to (1 - i)^2:a is 1, b is i. Then:(1)^2 - 2*(1)*(i) + (i)^2Calculating each term:1 squared is 1.2*1*i is 2i, so with the negative sign, it's -2i.i squared is -1.So putting them all together: 1 - 2i + (-1) = (1 -1) -2i = 0 -2i = -2i.Same result. Okay, that confirms it. So regardless of the method I use, expanding the product directly or using the algebraic identity for squaring a binomial, I end up with -2i. Therefore, (1 - i)^2 is equal to -2i.Final AnswerThe result of ((1 - i)^2) is boxed{-2i}.</think>To calculate ((1 - i)^2) where (i) is the imaginary unit, we can use the distributive property (FOIL method) or the algebraic identity for squaring a binomial.Using the distributive property:[(1 - i)^2 = (1 - i)(1 - i)]Expanding this product step by step:- First terms: (1 cdot 1 = 1)- Outer terms: (1 cdot (-i) = -i)- Inner terms: ((-i) cdot 1 = -i)- Last terms: ((-i) cdot (-i) = i^2 = -1)Adding all these terms together:[1 - i - i - 1 = (1 - 1) + (-i - i) = 0 - 2i = -2i]Using the algebraic identity ((a - b)^2 = a^2 - 2ab + b^2) where (a = 1) and (b = i):[(1 - i)^2 = 1^2 - 2 cdot 1 cdot i + i^2 = 1 - 2i + (-1) = (1 - 1) - 2i = 0 - 2i = -2i]Both methods confirm that the result is (boxed{-2i}).]

🔑:(1-i)^2 = 1 - 2i + i^2 = -2i.Therefore, the answer is boxed{-2i}.

🧠:[<think>Okay, let's tackle these combinatorics problems step by step. I need to figure out the number of different ways to select and distribute the books under each of the three conditions. Let me start with the first one.Problem (1): Select 2 storybooks and 2 math books.Alright, so we have 5 different storybooks and 4 different math books. We need to choose 2 storybooks and 2 math books, then distribute these 4 books to 4 students, one each. First, I think the process involves two main steps: selecting the books and then arranging them among the students. Let me confirm that.For selecting 2 storybooks out of 5, the number of ways is combination of 5 taken 2 at a time, which is denoted as C(5,2). Similarly, for selecting 2 math books out of 4, it's C(4,2). Then, once we have selected the 4 books, we need to distribute them to 4 students. Since the books are different and each student gets one book, this is a permutation problem. So, 4 books can be arranged in 4! ways.Therefore, the total number of ways should be C(5,2) * C(4,2) * 4!.Let me compute that:C(5,2) = 5! / (2! * 3!) = (5*4)/2 = 10C(4,2) = 4! / (2! * 2!) = (4*3)/2 = 64! = 24Multiply them all: 10 * 6 * 24. Let's do 10*6=60, then 60*24=1440.So, the answer for part (1) is 1440. But wait, let me check if I considered the order correctly. Since we are distributing the books to different students, the order matters, which we accounted for by multiplying with 4!. So yes, that seems right. If we had just selected the books without distributing, it would be 10*6=60, but since each distribution is different, multiplying by 24 is correct. Okay, that makes sense.Problem (2): Ensure one specific storybook and one specific math book are among the selected.Hmm. This requires that a specific storybook (let's say Storybook A) and a specific math book (Mathbook B) must be included in the 4 books selected. Then distribute them to the students.So, since one storybook and one math book are already chosen, we need to choose the remaining 2 books from the remaining storybooks and math books. However, we need to make sure that the total number of storybooks and math books meets the criteria. Wait, the problem says "Ensure one specific storybook and one specific math book are among the selected". It doesn't specify that exactly those two, but just that both of them must be included. So, the other two books can be any of the remaining 4 storybooks and 3 math books.Wait, original counts: 5 storybooks (including the specific one) and 4 math books (including the specific one). If we have to include the specific storybook and the specific math book, then we need to choose 3 more books? Wait, no. Wait, total books to select is 4. If we must include 1 specific storybook and 1 specific math book, then we need to choose 4 - 2 = 2 more books. But these two can be any combination of storybooks and math books, right? As long as we have at least the specific ones.But wait, the problem doesn't specify that we need exactly those two, but just that those two are included. So, the remaining two can be any of the remaining 4 storybooks and 3 math books. So, total remaining books: 4 + 3 = 7. But we need to choose 2 more books. However, there is a catch here. If we choose 2 more storybooks, that's okay. If we choose 1 storybook and 1 math book, that's also okay. If we choose 2 math books, that's okay too. So, all combinations are allowed as long as the total is 4 books with the specific story and math book included.Therefore, the number of ways to choose the remaining 2 books is C(4 + 3, 2) = C(7,2). Wait, but this would include all possibilities. However, the problem is that the original problem statement might have restrictions. Wait, no. Since we need to include at least 1 specific storybook and 1 specific math book, but the rest can be any. So, the remaining 2 books are selected from the remaining 4 storybooks and 3 math books. So, 7 books in total, choose 2. So C(7,2). Then, the total number of ways is C(7,2). Then, multiplied by the number of ways to arrange the 4 books (including the specific ones) to the 4 students. But wait, similar to part (1), once the books are selected, we have to permute them.But hold on, the specific storybook and math book are already included, so after choosing the remaining 2 books, we have 4 books in total. Then, distribute them to 4 students, which is 4! ways. Therefore, the total number of ways is C(7,2) * 4!.Wait, let me check: 7 choose 2 is 21, multiplied by 24 (which is 4!) is 504. So 21 * 24 = 504. Is that the answer? Wait, but let me make sure. Alternatively, maybe I need to break it down more carefully.Another way: The total number of ways to choose 4 books that include the specific storybook and the specific math book. So, since we have to include those two, we need to choose 2 more books from the remaining 4 storybooks and 3 math books. So, that is C(4 + 3, 2) = C(7,2) = 21. Then, for each such selection, we have 4 distinct books, which can be arranged in 4! = 24 ways. So, total is 21 * 24 = 504. That seems correct.But wait, another approach: Alternatively, think of the problem as first selecting the specific storybook and specific math book, then choosing 2 more books, then permuting. So that's 1 (specific story) * 1 (specific math) * C(7,2) * 4!.Which is the same as above. So yes, 504. So the answer is 504.But wait, wait, maybe another way. Let me check. Suppose instead of combining all, we consider different cases for the remaining two books. Like, case 1: two storybooks, case 2: one storybook and one math, case 3: two math books.But perhaps that's overcomplicating, but let's try.Case 1: Select 2 more storybooks. Since we already have 1 storybook and 1 math book. So total storybooks: 3, math:1. Wait, but in the original problem, there's no restriction on the number of story or math books. The only restriction is that specific ones are included. So, actually, the remaining two can be any combination. But perhaps splitting into cases is useful.Case 1: 2 storybooks. So, the total number of storybooks selected is 1 (specific) + 2 = 3. The math books are 1 (specific) + 0 = 1. So, selecting 2 storybooks from remaining 4. Then the number of ways: C(4,2) * C(3,0) [for math]. Then, arrange all 4 books.Similarly, Case 2: 1 storybook and 1 math book. Then total storybooks: 1 + 1 = 2, math:1 + 1 = 2. Number of ways: C(4,1)*C(3,1).Case 3: 2 math books. Storybooks:1 + 0 =1, math:1 + 2 =3. Number of ways: C(4,0)*C(3,2).Then, sum all cases and multiply by 4!.So let's compute:Case 1: C(4,2)*C(3,0) = 6*1=6Case 2: C(4,1)*C(3,1)=4*3=12Case 3: C(4,0)*C(3,2)=1*3=3Total ways to choose: 6 + 12 + 3 = 21. Then multiply by 4! =24. So 21*24=504. Same as before. So this confirms the answer. So whether we do it in cases or not, we get the same result. So, that's a good check. Therefore, 504 is correct.Problem (3): Ensure at least 3 of the selected books are storybooks.Okay, so we need to select 4 books with at least 3 storybooks. Since there are 5 storybooks and 4 math books.So "at least 3 storybooks" means either exactly 3 storybooks and 1 math book, or exactly 4 storybooks and 0 math books.Therefore, we have two cases:Case 1: 3 storybooks, 1 math book.Case 2: 4 storybooks, 0 math books.We need to compute the number of ways for each case, then sum them, then multiply by 4! (since we need to distribute the books to the students).So first, compute the combinations for each case.Case 1: C(5,3) * C(4,1)Case 2: C(5,4) * C(4,0)Then, total number of ways is [C(5,3)*C(4,1) + C(5,4)*C(4,0)] * 4!Compute each term:C(5,3) = 10C(4,1) =4So Case 1: 10*4=40Case 2:C(5,4)=5C(4,0)=1Case 2:5*1=5Total combinations:40 +5=45Multiply by 4! =24. So 45*24.Let me compute that: 40*24=960, 5*24=120, total 960 +120=1080.Therefore, the answer is 1080.But let me verify this approach.Alternatively, some people might think of subtracting the cases where fewer than 3 storybooks are selected from the total. But since the problem is about selecting and distributing, it's better to compute directly.But let's confirm with another method. The total number of ways to choose any 4 books and distribute them is C(5+4,4)*4! But wait, no. Wait, actually, when selecting books, since they are different, and the order of selection matters when distributing. Wait, actually, the total number of ways is (number of ways to choose 4 books) * (number of permutations). However, in reality, the total number of ways is P(9,4) = 9*8*7*6, which is 3024. But let's see if our previous answer of 1080 is plausible.Alternatively, let's compute the total number of ways where at least 3 storybooks are selected. So, as we did before.Case 1: 3 story,1 math. Number of ways to choose: C(5,3)*C(4,1)=10*4=40. Then, permute:40*24=960.Case 2:4 story,0 math. C(5,4)*C(4,0)=5*1=5. Then permute:5*24=120. Total:960+120=1080. Which is the same as before. So that seems correct.Alternatively, if I compute the total number of ways to choose and distribute any 4 books: 9 books total, choose 4, then arrange. So, C(9,4)*4! = 126 *24=3024. Then, the number of ways with at least 3 storybooks is 1080, which is less than half of 3024. That seems plausible. Let me check if 1080 is indeed correct.Alternatively, compute the number of ways with exactly 3 storybooks and 1 math book:C(5,3)*C(4,1)=10*4=40. Then, 40 selections, each of 4 books. Then, distributing them is 4! ways. So 40*24=960.Similarly, exactly 4 storybooks: C(5,4)*C(4,0)=5*1=5. Then, 5*24=120. So total 960+120=1080. Yes, that's correct. So the answer is 1080.Wait, but let me make sure that when we compute C(5,3)*C(4,1), we are considering the different combinations. Since all the books are different, so yes, each combination is unique, and then permuting them gives unique ordered arrangements. So that's correct.Therefore, all three answers are:1) 14402) 5043) 1080Final Answer(1) boxed{1440} (2) boxed{504} (3) boxed{1080}</think>To solve the given problems, we need to consider the combinations of selecting books and the permutations for distributing them to students. Here are the detailed steps:(1) Select 2 storybooks and 2 math books:1. Calculate the number of ways to choose 2 storybooks from 5: ( binom{5}{2} = 10 ).2. Calculate the number of ways to choose 2 math books from 4: ( binom{4}{2} = 6 ).3. Calculate the number of ways to distribute 4 books to 4 students: ( 4! = 24 ).4. Multiply these values together: ( 10 times 6 times 24 = 1440 ).(2) Ensure one specific storybook and one specific math book are among the selected:1. After selecting the specific storybook and math book, we need to choose 2 more books from the remaining 4 storybooks and 3 math books: ( binom{7}{2} = 21 ).2. Calculate the number of ways to distribute 4 books to 4 students: ( 4! = 24 ).3. Multiply these values together: ( 21 times 24 = 504 ).(3) Ensure at least 3 of the selected books are storybooks:1. Consider two cases: 3 storybooks and 1 math book, or 4 storybooks and 0 math books.2. For 3 storybooks and 1 math book: ( binom{5}{3} times binom{4}{1} = 10 times 4 = 40 ).3. For 4 storybooks and 0 math books: ( binom{5}{4} times binom{4}{0} = 5 times 1 = 5 ).4. Sum the combinations: ( 40 + 5 = 45 ).5. Calculate the number of ways to distribute 4 books to 4 students: ( 4! = 24 ).6. Multiply these values together: ( 45 times 24 = 1080 ). Final Answers(1) boxed{1440} (2) boxed{504} (3) boxed{1080},<think>Okay, let's tackle these problems one by one. I need to make sure I understand each part correctly and then figure out the right approach. Let's start with question (1).Problem (1): Select 2 storybooks and 2 math books.So, we have 5 different storybooks and 4 different math books. We need to choose 2 storybooks and 2 math books, and then give these 4 books to 4 students, one each. Wait, actually, the question says "select 4 books and given to 4 students, one book per student." So first, we need to select the books and then distribute them. But the problem is asking for the number of different ways to do both steps.Hmm. So first, we select the books and then arrange them among the students. So the total number of ways would be the number of ways to choose the books multiplied by the number of ways to distribute them to the students.For part (1), selecting 2 storybooks and 2 math books. The number of ways to choose 2 storybooks from 5 is combination of 5 taken 2 at a time, which is C(5,2). Similarly, the number of ways to choose 2 math books from 4 is C(4,2). Then, once we've selected these 4 books, we need to distribute them to 4 students. Since the books are different and the students are distinct, the number of ways to distribute them would be 4! (since each of the 4 books can go to any of the 4 students).So the total number of ways is C(5,2) * C(4,2) * 4!.Let me compute that. C(5,2) is 10, C(4,2) is 6, and 4! is 24. So 10*6=60, 60*24=1440. So 1440 ways.Wait, but let me double-check. So when we choose 2 storybooks and 2 math books, and then arrange all 4 books to 4 students, yes, that's correct. Because the selection is without order, but the distribution is with order. So yes, multiply by 4!.Okay, that seems right.Problem (2): Ensure one specific storybook and one specific math book are among the selected.So here, there's a specific storybook, say S1, and a specific math book, say M1, that must be included in the selected books. So out of the 4 books given to the students, S1 and M1 must be there. Then, the remaining 2 books can be any from the remaining storybooks and math books.So first, we need to select S1 and M1. Then, we need to choose 2 more books from the remaining 4 storybooks (since one storybook is already chosen) and 3 math books (since one math book is already chosen). But wait, the remaining 2 books can be any combination, right? So they can be 0 storybooks and 2 math books, 1 storybook and 1 math book, or 2 storybooks and 0 math books.But since the total number of books selected is 4, with 2 already being specific ones, we need 2 more. So the remaining two can be from 4 storybooks and 3 math books. So the number of ways to choose the remaining 2 books is C(4 + 3, 2) = C(7,2) = 21.But wait, but in this case, are we allowed to pick more math books? The original problem says "from 5 different storybooks and 4 different math books," so when we have to include one specific storybook and one specific math book, the remaining two can be any of the remaining 4 storybooks and 3 math books. Therefore, yes, the number of ways is C(7,2).But then, once we have selected those 4 books (S1, M1, and 2 others), we need to distribute them to the 4 students. So again, the number of arrangements is 4!.But wait, hold on. Alternatively, is the selection process different? Since the problem states "ensure one specific storybook and one specific math book are among the selected." So we have to include those two specific books. So the selection is 1 specific storybook, 1 specific math book, and then 2 more books from the remaining 4 storybooks and 3 math books.Therefore, the number of ways is C(1,1)*C(1,1)*C(7,2)*4!.But C(1,1) is 1, so it's 1*1*21*24 = 504.Wait, but let me check again. Let me think step by step.First, selecting the books. Must include S1 and M1. So the remaining two books can be any of the remaining 4 storybooks and 3 math books. So the number of ways to choose those two is C(4 + 3, 2) = 21. Then, once we have the 4 books, we need to distribute them to 4 students, which is 4! permutations. So total is 21*24=504.Alternatively, another way: the total number of ways to select 4 books that include S1 and M1. So the number of such selections is C(7,2) as we need to choose 2 more from 7 remaining books. Then, for each such selection, the number of ways to distribute is 4!.Alternatively, another approach: fix S1 and M1 as being selected. Then choose 2 more books from the remaining 7 (4 story + 3 math). Then arrange all 4 books. So yes, same as before.Therefore, 21*24=504. So answer is 504.Wait, but let me confirm once more. Suppose S1 is one of the storybooks and M1 is one of the math books. So to ensure they are included, we have to include them. So when choosing the books, S1 is already in, M1 is already in. Then, from the remaining 5-1=4 storybooks and 4-1=3 math books, we choose 2. So 4+3=7 books, choose 2. So 7 choose 2. Then multiply by 4! for distributing. Yes, so 21*24=504. Okay, that seems correct.Problem (3): Ensure at least 3 of the selected books are storybooks.So here, we need to select 4 books where at least 3 are storybooks. That means either exactly 3 storybooks and 1 math book, or exactly 4 storybooks and 0 math books.So we need to compute both cases and add them together.First, case 1: 3 storybooks and 1 math book.Number of ways to choose the books: C(5,3)*C(4,1). Then, distribute the 4 books to the students: 4!.Case 2: 4 storybooks and 0 math books.Number of ways to choose the books: C(5,4)*C(4,0). Then distribute them: 4!.Then total is [C(5,3)*C(4,1) + C(5,4)*C(4,0)] *4!.Compute:C(5,3)=10, C(4,1)=4. So 10*4=40.C(5,4)=5, C(4,0)=1. So 5*1=5.Total books selections: 40 +5=45.Multiply by 4! =24. So 45*24.Compute 45*24: 45*20=900, 45*4=180, total 1080.So answer is 1080.Wait, let me confirm.First, selecting 3 storybooks from 5: C(5,3)=10. Then 1 math from 4: C(4,1)=4. Multiply: 10*4=40.Then selecting 4 storybooks from 5: C(5,4)=5. 0 math books: C(4,0)=1. Multiply:5*1=5.Total selections:40+5=45. Then for each selection, arrange the 4 books to 4 students:4! =24. So total ways:45*24=1080.Yes, that seems correct.Alternatively, another way to think: Total number of ways without any restriction is C(9,4)*4! = 126*24=3024. But that's not needed here since we have to consider only the cases where at least 3 are storybooks.But since we did the calculation directly, I think 1080 is correct.Let me double-check the numbers:C(5,3)=10, C(4,1)=4, so 10*4=40. C(5,4)=5, so total 45. 45*24=1080. Yes.Therefore, answers are:(1) 1440(2) 504(3) 1080Final Answer(1) boxed{1440} (2) boxed{504} (3) boxed{1080}</think>To solve the given problems, we need to consider both the selection of books and their distribution to the students. Here are the detailed steps:(1) Select 2 storybooks and 2 math books:1. Selection: Choose 2 storybooks from 5 and 2 math books from 4. - Number of ways to choose storybooks: ( binom{5}{2} = 10 ) - Number of ways to choose math books: ( binom{4}{2} = 6 )2. Distribution: Distribute the 4 selected books to 4 students. - Number of ways to distribute: ( 4! = 24 )3. Total: Multiply the selection and distribution results. - ( 10 times 6 times 24 = 1440 )(2) Ensure one specific storybook and one specific math book are among the selected:1. Selection: The specific storybook and math book are already chosen. Choose the remaining 2 books from the remaining 4 storybooks and 3 math books. - Number of ways to choose remaining books: ( binom{4 + 3}{2} = binom{7}{2} = 21 )2. Distribution: Distribute the 4 selected books to 4 students. - Number of ways to distribute: ( 4! = 24 )3. Total: Multiply the selection and distribution results. - ( 21 times 24 = 504 )(3) Ensure at least 3 of the selected books are storybooks:1. Case 1: 3 storybooks and 1 math book - Number of ways to choose storybooks: ( binom{5}{3} = 10 ) - Number of ways to choose math books: ( binom{4}{1} = 4 ) - Total for this case: ( 10 times 4 = 40 )2. Case 2: 4 storybooks and 0 math books - Number of ways to choose storybooks: ( binom{5}{4} = 5 ) - Number of ways to choose math books: ( binom{4}{0} = 1 ) - Total for this case: ( 5 times 1 = 5 )3. Total Selection: Combine both cases. - ( 40 + 5 = 45 )4. Distribution: Distribute the 4 selected books to 4 students. - Number of ways to distribute: ( 4! = 24 )5. Total: Multiply the total selection and distribution results. - ( 45 times 24 = 1080 ) Final Answers(1) (boxed{1440}) (2) (boxed{504}) (3) (boxed{1080})]

🔑:Solution: (1) According to the problem, first select 2 from the 5 different storybooks and 2 from the 4 different math books, then distribute them to the 4 students. We get C_{5}^{2} C_{4}^{2} A_{4}^{4} = 1440…(4 points) (2) If one specific storybook and one specific math book must be selected, then choose 2 from the remaining 7 books, and distribute them to the 4 students. We get C_{7}^{2} A_{4}^{4} = 504…(9 points) (3) If at least 3 of the selected books are storybooks, including scenarios with 3 storybooks and 1 math book, or 4 storybooks, we get C_{5}^{3} C_{4}^{1} A_{4}^{4} + A_{5}^{4} = 1080…(14 points)Therefore, the answers are: (1) boxed{1440} (2) boxed{504} (3) boxed{1080}

🧠:[<think>Okay, let me try to figure out this problem step by step. So, we have circle C with the equation (x + 2)^2 + y^2 = 4. That means its center is at (-2, 0) and the radius is 2, right? Because the standard form is (x - h)^2 + (y - k)^2 = r^2. So, h is -2, k is 0, so center at (-2, 0), radius squared is 4, so radius 2. Got that.Then, there are two perpendicular lines, l1 and l2, both passing through point A(2, 0). So, point A is at (2, 0). The lines are perpendicular, so they form a right angle where they intersect. But since they both pass through A(2, 0), they must intersect at that point. So, l1 and l2 are two perpendicular lines intersecting at A(2, 0). So, they could be, for example, horizontal and vertical lines, but since they are perpendicular, they could have slopes that are negative reciprocals. For instance, one could have slope m and the other -1/m.Then, we have a circle with center M(1, m), where m > 0. So, the center is at (1, m), and m is positive. This circle is tangent to circle C and also tangent to both lines l1 and l2. We need to find the equation of circle M.First, let's recall that if two circles are tangent to each other, the distance between their centers is equal to the sum or difference of their radii. Since they can be externally tangent or internally tangent. Since circle C has radius 2, and circle M is another circle. If circle M is outside circle C, then the distance between centers would be the sum of radii. If it's inside, then the difference. But given that the center of M is at (1, m), which is to the right of circle C's center (-2, 0), so the distance between (-2,0) and (1, m) is sqrt[(1 - (-2))^2 + (m - 0)^2] = sqrt[9 + m²]. Now, circle C has radius 2, and circle M has radius r. If they are tangent, then sqrt(9 + m²) = 2 + r or |2 - r|. But since the center M is at (1, m), which is 3 units away from (-2,0) in the x-direction and m units in the y-direction. Let's see. If the circles are tangent externally, then the distance between centers is sum of radii, if internally, then the difference.But let's also consider that circle M is tangent to both lines l1 and l2. So, the distance from center M(1, m) to each of these lines should be equal to its radius r. Since the circle is tangent to both lines, the radius is the perpendicular distance from the center to each line. Also, since l1 and l2 are perpendicular, and both pass through A(2,0), the lines form a sort of right angle at point A. So, the circle M is tangent to both lines, which are perpendicular, so the center M(1, m) must lie at a distance r from both lines. Hmm, also, since the lines are perpendicular, the distance from M to both lines is equal to the radius. That might mean that M is equidistant from both lines, but given the lines are perpendicular, maybe the radius is related to the distance from M to the point A(2,0)?Wait, not necessarily. Wait, if you have two perpendicular lines intersecting at a point, then the set of points equidistant from both lines would lie on the angle bisectors, which would be the lines y = x and y = -x if the original lines are the axes. But in our case, the lines l1 and l2 are perpendicular but not necessarily the coordinate axes. Since they pass through (2,0), and are perpendicular, they could have any orientation. So, we need to find equations for l1 and l2 such that they are perpendicular and pass through (2,0), but since the problem doesn't specify their exact equations, we might need to consider general cases. But since the circle M is tangent to both lines, and the lines are perpendicular, perhaps we can parametrize the lines?Alternatively, maybe there's a way to relate the distance from M(1, m) to both lines l1 and l2 as equal to the radius r. Since both distances are equal to r, and the lines are perpendicular, we can use some properties here.Also, the circle M is tangent to circle C. So, we have two conditions: the distance between M and C is equal to 2 ± r (depending on external or internal tangent), and the distance from M to both lines l1 and l2 is equal to r.But since the problem doesn't specify if the tangent is external or internal, we might need to consider both possibilities. However, given that the center M is at (1, m), which is 3 units away from the center of circle C at (-2, 0) in the x-direction. If the radius r is such that the distance between centers is 2 + r, then sqrt(9 + m²) = 2 + r. If it's 2 - r, then sqrt(9 + m²) = |2 - r|. But since m > 0 and r must be positive, we need to check which case is possible.But let's think about the position. If circle M is outside circle C, then sqrt(9 + m²) = 2 + r. If it's inside, then sqrt(9 + m²) = |2 - r|. But since center M is at (1, m), which is 3 units to the right of (-2, 0). The radius of circle C is 2, so the rightmost point of circle C is at (-2 + 2, 0) = (0, 0). So, center M is at (1, m), which is outside circle C. Therefore, the distance between centers must be equal to the sum of radii if they are externally tangent. Therefore, sqrt(9 + m²) = 2 + r.Alternatively, if circle M were inside circle C, but since center M is at (1, m), which is 1 unit right of the origin, and circle C is centered at (-2,0) with radius 2, the distance from M to C's center is sqrt(9 + m²), which is at least 3 (since m >=0). But circle C's radius is 2, so for M to be inside C, sqrt(9 + m²) + r <= 2? But sqrt(9 + m²) is already at least 3, which is larger than 2, so that's impossible. Therefore, the circles must be externally tangent, so sqrt(9 + m²) = 2 + r.So, that gives us one equation: sqrt(9 + m²) = 2 + r. So, r = sqrt(9 + m²) - 2.Now, the other condition is that the circle M is tangent to both lines l1 and l2. So, the distance from M(1, m) to each line is equal to the radius r. Since both lines pass through point A(2, 0), and are perpendicular, we need to find the equations of these lines such that the distance from M to each line is r.But since the lines are not given, we need to parametrize them. Let's denote the lines as l1 and l2. Since they are perpendicular and pass through (2, 0), we can represent them in the form:Let’s assume l1 has a slope of k, then l2, being perpendicular, will have a slope of -1/k (since the product of slopes of perpendicular lines is -1). However, if one of the lines is vertical, the other would be horizontal. But since vertical lines have undefined slopes, we might need to consider that case separately. However, for generality, let's suppose that neither line is vertical or horizontal. Then, we can write the equations of l1 and l2 as:For l1: y - 0 = k(x - 2) => y = kx - 2kFor l2: y - 0 = (-1/k)(x - 2) => y = (-1/k)x + 2/kNow, the distance from point M(1, m) to line l1 is equal to r, and similarly the distance to l2 is also equal to r.The formula for the distance from a point (x0, y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a² + b²).So, let's write the equations of l1 and l2 in ax + by + c = 0 form.For l1: y = kx - 2k => kx - y - 2k = 0So, a = k, b = -1, c = -2kDistance from M(1, m) to l1 is |k*1 + (-1)*m + (-2k)| / sqrt(k² + 1) = |k - m - 2k| / sqrt(k² + 1) = |-k - m| / sqrt(k² + 1) = |k + m| / sqrt(k² + 1) (since absolute value is always positive)Similarly, for l2: y = (-1/k)x + 2/k => (1/k)x + y - 2/k = 0Multiply both sides by k to eliminate the denominator (but since we are writing in ax + by + c = 0, we can also write it as x + ky - 2 = 0 (multiplying numerator and denominator by k, but assuming k ≠ 0).Wait, actually, let's do it properly. Starting from y = (-1/k)x + 2/k, rearranged:(1/k)x + y - 2/k = 0But to make it easier, multiply both sides by k (assuming k ≠ 0):x + ky - 2 = 0So, the equation of l2 is x + ky - 2 = 0, so a = 1, b = k, c = -2Then, the distance from M(1, m) to l2 is |1*1 + k*m - 2| / sqrt(1² + k²) = |1 + km - 2| / sqrt(1 + k²) = |km - 1| / sqrt(1 + k²)Since both distances must be equal to r, we have:|k + m| / sqrt(k² + 1) = r|km - 1| / sqrt(k² + 1) = rTherefore, we have two equations:1) |k + m| = r * sqrt(k² + 1)2) |km - 1| = r * sqrt(k² + 1)Since both left sides equal the same right side, we can set them equal to each other:|k + m| = |km - 1|Also, remember that r = sqrt(9 + m²) - 2 from the earlier condition.But this seems a bit complicated. Let me think. Since both equations equal r times sqrt(k² +1), then |k + m| = |km -1|So, either k + m = km -1 or k + m = -(km -1)Case 1: k + m = km -1Then, rearranged: km - k - m -1 =0Factor: k(m -1) - (m +1) =0Not sure. Alternatively, add 1 to both sides: km -k -m =1Hmm, maybe factor:km -k -m =1km -k -m +1 = 2(k -1)(m -1) =2Wait, let's see:km -k -m =1Add 1 to both sides:km -k -m +1 =2Which factors as (k -1)(m -1)=2Yes, because (k -1)(m -1) = km -k -m +1 =2So, (k -1)(m -1) =2Case 2: k + m = -km +1Then, k + m + km -1 =0Factor: k(1 + m) + (m -1)=0Not sure. Alternatively:km + k + m -1 =0Which can be written as (k +1)(m +1) -2 =0Wait:km +k +m -1 = (k +1)(m +1) -2 =0Because (k +1)(m +1) = km +k +m +1, so subtract 2 gives km +k +m -1Therefore, (k +1)(m +1) =2So, in Case 1: (k -1)(m -1)=2Case 2: (k +1)(m +1)=2So, these are the two possibilities.Now, we need to relate this with the other equation, which is r = sqrt(9 + m²) - 2, and also r = |k + m| / sqrt(k² +1) (from the first distance equation). Similarly, since both distances equal r, we can use either one. Let's pick the first one.So, r = |k + m| / sqrt(k² +1)But also r = sqrt(9 + m²) - 2Therefore, |k + m| / sqrt(k² +1) = sqrt(9 + m²) - 2But since m >0 and we don't know the sign of k + m, but given that the lines are passing through (2,0), and the center M is at (1, m), m >0, maybe we can assume that k + m is positive? Not sure. Maybe need to consider both cases.Alternatively, since (k -1)(m -1) =2 or (k +1)(m +1)=2, perhaps we can solve for k in terms of m and substitute into the equation involving r.Let's try for Case 1: (k -1)(m -1)=2Solve for k:k -1 = 2 / (m -1)So, k = 1 + 2/(m -1)Similarly, in Case 2: (k +1)(m +1)=2Solve for k:k +1 = 2/(m +1)So, k = (2/(m +1)) -1 = (2 - (m +1))/(m +1)) = (1 - m)/(m +1)So, we can substitute these expressions for k into the equation |k + m| / sqrt(k² +1) = sqrt(9 + m²) - 2But since in each case, we have a specific expression for k, let's substitute those.Starting with Case 1: k =1 + 2/(m -1)Let’s compute k + m:k + m =1 + 2/(m -1) + m = m +1 + 2/(m -1)Similarly, sqrt(k² +1):k² +1 = [1 + 2/(m -1)]² +1 = 1 + 4/(m -1) + 4/(m -1)^2 +1 = 2 + 4/(m -1) + 4/(m -1)^2So, sqrt(k² +1) = sqrt(2 + 4/(m -1) + 4/(m -1)^2)Hmm, this might get complicated, but let's see.So, the left side is |k + m| / sqrt(k² +1) = [m +1 + 2/(m -1)] / sqrt(2 + 4/(m -1) + 4/(m -1)^2)The right side is sqrt(9 + m²) - 2This seems messy, but maybe we can simplify.Alternatively, perhaps let’s set t = m -1, so m = t +1. Then, k =1 + 2/tBut m >0, so t +1 >0 => t > -1. Since m >0, and m = t +1, so t = m -1. Since m is positive, t can be greater than -1. But maybe t is positive? Not sure. Let's see.Alternatively, maybe try to find m such that the equation holds.Alternatively, maybe Case 2 is simpler.Case 2: k = (1 - m)/(m +1)Compute k + m:k + m = (1 - m)/(m +1) + m = [1 - m + m(m +1)] / (m +1) = [1 - m + m² + m]/(m +1) = (1 + m²)/(m +1)Similarly, sqrt(k² +1):k² +1 = [(1 - m)^2 / (m +1)^2] +1 = [ (1 - 2m + m²) + (m +1)^2 ] / (m +1)^2Wait, no. Let me compute it step by step.k² +1 = [(1 - m)/(m +1)]² +1 = (1 - 2m + m²)/(m +1)^2 +1 = (1 - 2m + m² + (m +1)^2 ) / (m +1)^2Compute numerator:1 -2m + m² + m² +2m +1 = 1 -2m + m² + m² +2m +1 = 2 + 2m²So, numerator is 2 + 2m², denominator is (m +1)^2Therefore, sqrt(k² +1) = sqrt( (2 + 2m²)/(m +1)^2 ) = sqrt(2(1 + m²))/|m +1|Since m >0, m +1 >0, so |m +1| = m +1. Therefore, sqrt(2(1 + m²))/(m +1)So, the left side is |k + m| / sqrt(k² +1) = [ (1 + m²)/(m +1) ] / [ sqrt(2(1 + m²))/(m +1) ) ] = [ (1 + m²)/(m +1) ] * [ (m +1)/sqrt(2(1 + m²)) ) ] = (1 + m²)/sqrt(2(1 + m²)) = sqrt(1 + m²)/sqrt(2) = sqrt( (1 + m²)/2 )Therefore, left side is sqrt( (1 + m²)/2 )Right side is sqrt(9 + m²) - 2Therefore, equation:sqrt( (1 + m²)/2 ) = sqrt(9 + m²) - 2Let me square both sides to eliminate the square roots. But first, let me denote:Let’s set left side = sqrt( (1 + m²)/2 )Right side = sqrt(9 + m²) - 2So, sqrt( (1 + m²)/2 ) + 2 = sqrt(9 + m²)Wait, but if we move the 2 to the left side:sqrt(9 + m²) = sqrt( (1 + m²)/2 ) + 2Then square both sides:9 + m² = [ sqrt( (1 + m²)/2 ) + 2 ]^2 = ( (1 + m²)/2 ) + 4 + 4*sqrt( (1 + m²)/2 )Simplify the right side:(1 + m²)/2 + 4 + 4*sqrt( (1 + m²)/2 ) = (1 + m²)/2 + 8/2 + 4*sqrt( (1 + m²)/2 ) = (9 + m²)/2 + 4*sqrt( (1 + m²)/2 )So, equation becomes:9 + m² = (9 + m²)/2 + 4*sqrt( (1 + m²)/2 )Multiply both sides by 2:2*(9 + m²) = (9 + m²) + 8*sqrt( (1 + m²)/2 )Simplify left side minus right side:2*(9 + m²) - (9 + m²) = 8*sqrt( (1 + m²)/2 )So,9 + m² = 8*sqrt( (1 + m²)/2 )Divide both sides by 8:(9 + m²)/8 = sqrt( (1 + m²)/2 )Square both sides again:( (9 + m²)/8 )² = (1 + m²)/2Multiply both sides by 64:(9 + m²)^2 = 32*(1 + m²)Expand left side:81 + 18m² + m^4 = 32 + 32m²Bring all terms to left side:81 + 18m² + m^4 -32 -32m² =0Simplify:49 -14m² + m^4 =0Which is:m^4 -14m² +49 =0This is a quadratic in terms of m². Let’s set u = m²:u² -14u +49=0Solve:u = [14 ± sqrt(196 -196)]/2 = [14 ±0]/2 =7Therefore, u=7, so m²=7 => m=√7 (since m>0)Thus, m=√7Then, check if this satisfies the original equation.Let’s compute sqrt( (1 +7)/2 ) = sqrt(8/2)=sqrt(4)=2Right side: sqrt(9 +7) -2=sqrt(16)-2=4-2=2Yes, 2=2. So, valid.Therefore, in Case 2, we get m=√7. Then, let's check Case 1.Case 1: (k -1)(m -1)=2If we proceed similarly, perhaps we get another solution. Let me try.In Case 1: k =1 + 2/(m -1)Then, let's compute the left side |k + m| / sqrt(k² +1) and set it equal to sqrt(9 + m²) -2First, compute k + m =1 + 2/(m -1) + m = m +1 + 2/(m -1)Let’s write this as (m +1) + 2/(m -1)Hmm, this seems more complicated, but let's try with m=√7.Wait, but since in Case 2 we already found m=√7, perhaps this is the only solution, but we need to check if Case 1 can yield another solution.But let's proceed.Let’s denote the left side as |k + m| / sqrt(k² +1) = |(m +1) + 2/(m -1)| / sqrt( [1 + 2/(m -1)]² +1 )But this seems complex, maybe substitute m=√7 into Case 1.If m=√7, then (k -1)(√7 -1)=2 => k -1=2/(√7 -1)= [2(√7 +1)]/( (√7)^2 -1 )= [2(√7 +1)]/(7 -1)= [2(√7 +1)]/6= (√7 +1)/3Thus, k=1 + (√7 +1)/3= (3 +√7 +1)/3=(4 +√7)/3Then, let's compute the left side |k + m| / sqrt(k² +1 )k + m=(4 +√7)/3 +√7=(4 +√7 +3√7)/3=(4 +4√7)/3Absolute value is positive, so |k + m|=(4 +4√7)/3sqrt(k² +1):k=(4 +√7)/3, so k²= (16 +8√7 +7)/9=(23 +8√7)/9k² +1=(23 +8√7)/9 +9/9=(32 +8√7)/9sqrt(k² +1)=sqrt(32 +8√7)/3So, left side= (4 +4√7)/3 / [ sqrt(32 +8√7)/3 ]= (4 +4√7)/sqrt(32 +8√7 )Factor numerator and denominator:Numerator:4(1 +√7)Denominator: sqrt(8*(4 +√7))=sqrt(8)*sqrt(4 +√7)=2*sqrt(2)*sqrt(4 +√7)So, left side=4(1 +√7)/[2*sqrt(2)*sqrt(4 +√7)]=2(1 +√7)/[sqrt(2)*sqrt(4 +√7)]Multiply numerator and denominator by sqrt(2):=2(1 +√7)*sqrt(2)/[2*sqrt(4 +√7)]= (1 +√7)*sqrt(2)/sqrt(4 +√7)Now, let's square the numerator and denominator to see if they are equal:Numerator squared: (1 +√7)^2*2= (1 +2√7 +7)*2= (8 +2√7)*2=16 +4√7Denominator squared:4 +√7But (16 +4√7)/(4 +√7)= [4*(4 +√7)]/(4 +√7)=4So sqrt(16 +4√7)/sqrt(4 +√7)=sqrt(4)=2Therefore, left side= sqrt(16 +4√7)/sqrt(4 +√7)=2But the right side is sqrt(9 + (√7)^2) -2=sqrt(16)-2=4-2=2So, they are equal. Wait, but this was using m=√7 in Case 1. But how?Wait, but in Case 1, (k -1)(m -1)=2. When m=√7, we found that k=(4 +√7)/3, and substituting back into the equation gives that left side equals right side. But we already had m=√7 in Case 2. So, this suggests that m=√7 is a solution for both cases?Wait, but how is that possible? Because in Case 2, we had (k +1)(m +1)=2, leading to m=√7. But when we substituted m=√7 into Case 1, we also found a valid k. So, does that mean that for m=√7, both cases are possible? But that would mean there are two different lines l1 and l2 for the same m? But in reality, since the lines l1 and l2 are arbitrary perpendicular lines through A(2,0), there might be two different pairs of lines that satisfy the conditions for the same circle. But since the problem says "the equation of circle M is", it suggests there is a unique solution. Therefore, perhaps m=√7 is the only solution, and both cases lead to the same circle. Wait, but let's check.Wait, if m=√7, then the center is at (1, √7). The radius r= sqrt(9 + m²) -2= sqrt(9 +7)-2= sqrt(16)-2=4-2=2. So, radius is 2.Alternatively, in Case 1, we found k=(4 +√7)/3. Let's compute the distance from M(1,√7) to line l1: which is supposed to be r=2.Similarly, in Case 2, k=(1 -m)/(m +1)=(1 -√7)/(√7 +1). Let's rationalize:Multiply numerator and denominator by (√7 -1):[(1 -√7)(√7 -1)] / [ (√7 +1)(√7 -1) ] = [√7 -1 -7 +√7]/[7 -1] = [2√7 -8]/6 = (√7 -4)/3So, k=(√7 -4)/3Thus, in Case 2, k=(√7 -4)/3, which is negative since √7≈2.645, so √7 -4≈-1.355, so k≈-0.451Whereas in Case 1, k=(4 +√7)/3≈(4 +2.645)/3≈6.645/3≈2.215, which is positive.So, in Case 1, line l1 has a positive slope, and in Case 2, a negative slope. So, both pairs of lines (with positive and negative slopes) satisfy the conditions, but since the problem doesn't specify the lines, just that they are perpendicular through A(2,0), both are possible. However, the circle M is uniquely determined by its center (1, m) and radius r=2. Therefore, even though there are two different pairs of lines l1 and l2, the circle M is the same in both cases.Therefore, the equation of circle M is (x -1)^2 + (y -√7)^2 = (2)^2=4Wait, but let me confirm the radius. We had r= sqrt(9 + m²)-2. With m=√7, sqrt(9 +7)=sqrt(16)=4, so r=4-2=2. Correct.Alternatively, using the distance from M(1,√7) to line l1, which in Case 1 is |k + m| / sqrt(k² +1)=|(4 +√7)/3 +√7| / sqrt( ((4 +√7)/3)^2 +1 )But we already saw that this equals 2. So, yes, radius is 2.Therefore, the equation is (x -1)^2 + (y -√7)^2 =4But let me check if there's another possible solution from Case 1. Suppose there's another m value. But in Case 1, when we set up the equation, we ended up with a quartic equation which only had m=√7 as a solution. Wait, but in Case 1, we set up the equation sqrt( (1 + m²)/2 ) = sqrt(9 + m²) -2, leading to m=√7, and in Case 1, even though the algebra was more complex, substituting m=√7 also worked. So, maybe m=√7 is the only solution. Let me check.In Case 1, if we proceed with the algebra:We had the equation:sqrt( (1 + m²)/2 ) = sqrt(9 + m²) -2But we solved this for Case 2 and found m=√7. When we checked in Case 1, using m=√7 also worked. Therefore, perhaps there is only one solution, m=√7.Therefore, the equation of circle M is (x -1)^2 + (y -√7)^2=4But let me confirm with the original conditions:1. Distance between centers: sqrt( (1 +2)^2 + (√7 -0)^2 )=sqrt(9 +7)=sqrt(16)=4, which should be equal to 2 + r. Since r=2, 2 +2=4. Correct.2. The circle is tangent to both lines l1 and l2. Since we found the radius is 2, and the distance from M(1,√7) to each line is 2. Depending on the slope k, but regardless, since we derived that m=√7 satisfies the equation, it's valid.Therefore, the answer should be (x -1)^2 + (y -√7)^2=4But to confirm, let's check if there could be another solution. When we solved Case 2, we arrived at m=√7. In Case 1, when we substituted m=√7, it worked. But when solving Case 1's equation, we might get another solution. Wait, no. Let's recall that when we derived the quartic equation for Case 2, we arrived at m^4 -14m² +49=0, which factors as (m² -7)^2=0, so m²=7, so m=√7 only. Therefore, that equation has a double root at m²=7. Therefore, m=√7 is the only solution. Hence, no other solutions.Therefore, the equation is (x -1)^2 + (y -√7)^2=4Final AnswerThe equation of circle M is boxed{(x - 1)^2 + left(y - sqrt{7}right)^2 = 4}.</think>Given circle ( C ): ((x+2)^2 + y^2 = 4) with center ((-2, 0)) and radius 2, and two perpendicular lines ( l_1 ) and ( l_2 ) passing through point ( A(2, 0) ). We need to find the equation of circle ( M ) with center ( (1, m) ) (where ( m > 0 )) that is tangent to circle ( C ) and both lines ( l_1 ) and ( l_2 ).1. Distance between centers of circles ( C ) and ( M ): The distance between the centers ((-2, 0)) and ((1, m)) is (sqrt{(1 + 2)^2 + m^2} = sqrt{9 + m^2}). Since the circles are tangent, this distance equals the sum of their radii (externally tangent): [ sqrt{9 + m^2} = 2 + r ] where ( r ) is the radius of circle ( M ).2. Radius of circle ( M ): The radius ( r ) is also the perpendicular distance from the center ( (1, m) ) to the lines ( l_1 ) and ( l_2 ). For perpendicular lines passing through ( (2, 0) ), we derived the equations for the distances and found that: [ r = frac{|k + m|}{sqrt{k^2 + 1}} = frac{|km - 1|}{sqrt{k^2 + 1}} ] Solving these equations led to the conditions ((k - 1)(m - 1) = 2) or ((k + 1)(m + 1) = 2).3. Solving for ( m ): By substituting these conditions into the distance equations and solving, we found that ( m = sqrt{7} ) satisfies all conditions. The radius ( r ) was calculated as ( 2 ).4. Equation of circle ( M ): With center ( (1, sqrt{7}) ) and radius 2, the equation of circle ( M ) is: [ (x - 1)^2 + left(y - sqrt{7}right)^2 = 4 ]Thus, the equation of circle ( M ) is (boxed{(x - 1)^2 + left(y - sqrt{7}right)^2 = 4}).,<think>Okay, so I need to find the equation of circle M with center at (1, m) where m is positive. This circle has to be tangent to the given circle C: (x+2)^2 + y^2 = 4 and also tangent to two perpendicular lines l1 and l2 that both pass through point A(2, 0). Hmm, let me break this down step by step.First, let me visualize the problem. Circle C has its center at (-2, 0) and radius 2 because the standard form is (x - h)^2 + (y - k)^2 = r^2, so h is -2, k is 0, and r is 2. Then there's point A at (2, 0), which is 4 units to the right of the center of circle C on the x-axis. The two lines l1 and l2 are perpendicular and both pass through A(2, 0). So they form a kind of 'X' shape intersecting at A.Now, the circle M is centered at (1, m), which is somewhere to the right of circle C's center but still left of point A. Since m is positive, it's above the x-axis. This circle needs to be tangent to both lines l1 and l2 and also tangent to circle C. So, I need to find m such that all these conditions are satisfied.First, let me think about the tangent lines. Since l1 and l2 are perpendicular and pass through A(2, 0), their equations can be written in some general form. Let's denote the slopes of l1 and l2 as k and -1/k because they are perpendicular. But since they pass through (2, 0), their equations would be y = k(x - 2) and y = (-1/k)(x - 2). However, this might complicate things because we have two variables. Maybe there's a better way.Alternatively, since the two lines are perpendicular and intersect at A(2, 0), they can be represented as a pair of lines rotated by some angle θ. The distance from the center of circle M to each of these lines should be equal to the radius of circle M. Because the circle is tangent to both lines, the distance from the center (1, m) to each line must be equal to the radius r of circle M. Also, since the lines are perpendicular, the distances can be related through the angle between the lines.But maybe another approach is to recognize that the set of all lines passing through A(2, 0) that are perpendicular can be represented as a family of lines with varying slopes. However, since the circle M is tangent to both, perhaps the lines are symmetrical with respect to the x-axis? Wait, but point A is on the x-axis, and the center of circle M is at (1, m), which is above the x-axis. Maybe the lines are symmetrical in some way.Alternatively, since both lines pass through A(2, 0) and are perpendicular, the distance from the center M(1, m) to each line is equal to the radius. Let me denote the radius of circle M as r. So, if I can express the distance from (1, m) to each line l1 and l2 as r, and also the distance between the centers of circle C and M should be equal to the sum or difference of their radii (since they are tangent). Let me note that circle C has radius 2, and circle M has radius r. The distance between their centers is sqrt[(1 - (-2))^2 + (m - 0)^2] = sqrt[(3)^2 + m^2] = sqrt(9 + m^2). Since they are tangent, this distance should be equal to 2 + r (if externally tangent) or |2 - r| (if internally tangent). But since circle M is at (1, m) and circle C is at (-2, 0), and 2 + r would mean they don't overlap, but given that the center of M is outside circle C? Wait, let's check.The center of circle C is at (-2, 0) with radius 2. The center of circle M is at (1, m). The distance between centers is sqrt((1 + 2)^2 + (m - 0)^2) = sqrt(9 + m^2). Since sqrt(9 + m^2) is definitely greater than 2 (as m > 0, sqrt(9 + m^2) ≥ 3 > 2), so the circles must be externally tangent. Therefore, the distance between centers is equal to 2 + r. Therefore:sqrt(9 + m^2) = 2 + r. So, r = sqrt(9 + m^2) - 2. Equation (1)Now, for the lines l1 and l2. Since both lines pass through A(2, 0) and are perpendicular, the distance from M(1, m) to each line must be equal to r. Let's denote the two lines in general forms. Let me consider that any line passing through A(2, 0) can be written as a(x - 2) + b(y - 0) = 0, where a and b are coefficients. But since they are perpendicular, the product of their slopes is -1. Alternatively, maybe parametrize the lines by an angle.Let me consider that the two lines are at angles θ and θ + 90 degrees from the x-axis. Then their slopes would be tanθ and tan(θ + 90°) = -cotθ. So, equations of the lines would be y = tanθ(x - 2) and y = -cotθ(x - 2). Then, the distance from (1, m) to each line would be equal to r.The formula for the distance from a point (x0, y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a^2 + b^2). Let's write the lines in standard form.First line: y = tanθ(x - 2) => tanθ x - y - 2 tanθ = 0Second line: y = -cotθ(x - 2) => -cotθ x - y + 2 cotθ = 0So, the distance from (1, m) to the first line is |tanθ*1 - m - 2 tanθ| / sqrt(tan²θ + 1) = | - tanθ - m | / sqrt(tan²θ + 1)Similarly, distance to the second line is | -cotθ*1 - m + 2 cotθ | / sqrt(cot²θ + 1) = | cotθ - m | / sqrt(cot²θ + 1)But sqrt(tan²θ + 1) = secθ, and sqrt(cot²θ + 1) = cscθ. So the distances become:First distance: | - tanθ - m | / secθ = | - tanθ - m | * cosθ = | - sinθ - m cosθ |.Second distance: | cotθ - m | / cscθ = | cotθ - m | * sinθ = | cosθ - m sinθ |.Since both distances must equal r, we have:| - sinθ - m cosθ | = r,| cosθ - m sinθ | = r.So both expressions equal r, so they must be equal in absolute value. So:| - sinθ - m cosθ | = | cosθ - m sinθ |.Hmm, squaring both sides to eliminate the absolute value:(- sinθ - m cosθ)^2 = (cosθ - m sinθ)^2Expanding both sides:sin²θ + 2 m sinθ cosθ + m² cos²θ = cos²θ - 2 m cosθ sinθ + m² sin²θBring all terms to left side:sin²θ + 2 m sinθ cosθ + m² cos²θ - cos²θ + 2 m sinθ cosθ - m² sin²θ = 0Simplify term by term:sin²θ - cos²θ + (2m sinθ cosθ + 2m sinθ cosθ) + m² cos²θ - m² sin²θ = 0Which becomes:(sin²θ - cos²θ) + 4m sinθ cosθ + m² (cos²θ - sin²θ) = 0Factor terms:(1 - 2 cos²θ) + 4m sinθ cosθ + m² (cos2θ) = 0. Wait, since sin²θ - cos²θ = -cos2θ and cos²θ - sin²θ = cos2θ. Let me use double-angle identities.Wait, sin²θ - cos²θ = -cos2θ,4m sinθ cosθ = 2m sin2θ,and m² (cos²θ - sin²θ) = m² cos2θ.So substituting:- cos2θ + 2m sin2θ + m² cos2θ = 0Factor out cos2θ:cos2θ(-1 + m²) + 2m sin2θ = 0Rearranged:( m² - 1 ) cos2θ + 2m sin2θ = 0Let me divide both sides by cos2θ (assuming cos2θ ≠ 0):( m² - 1 ) + 2m tan2θ = 0So:tan2θ = (1 - m²) / (2m )Hmm, but θ is the angle of one of the lines. However, θ can vary, so perhaps this relationship must hold for some θ. But how do we relate this to the other condition, which involves the radius r and the tangency to circle C?From equation (1), we had r = sqrt(9 + m²) - 2.But also, from the distance to the lines, we have:r = | - sinθ - m cosθ | = | sinθ + m cosθ | (since absolute value) and r = | cosθ - m sinθ |.So, both | sinθ + m cosθ | and | cosθ - m sinθ | equal r. Let me call these equations (2) and (3).But since we have tan2θ = (1 - m²)/(2m), perhaps we can express sinθ and cosθ in terms of m. Let me set tan2θ = (1 - m²)/(2m). Let me denote tan2θ = T. So T = (1 - m²)/(2m). Then, since tan2θ = 2 tanθ / (1 - tan²θ), which is equal to T. Let me denote t = tanθ. Then:2t / (1 - t²) = (1 - m²)/(2m)Cross-multiplying:4mt = (1 - t²)(1 - m²)But this seems complicated. Alternatively, maybe express sinθ and cosθ in terms of m.Alternatively, note that from equations (2) and (3), | sinθ + m cosθ | = | cosθ - m sinθ |. Let me square both equations:(sinθ + m cosθ)^2 = r^2,(cosθ - m sinθ)^2 = r^2.Therefore:sin²θ + 2m sinθ cosθ + m² cos²θ = r²,cos²θ - 2m sinθ cosθ + m² sin²θ = r².Subtracting the two equations:[ sin²θ + 2m sinθ cosθ + m² cos²θ ] - [ cos²θ - 2m sinθ cosθ + m² sin²θ ] = 0Which simplifies to:sin²θ - cos²θ + 4m sinθ cosθ + m² (cos²θ - sin²θ ) = 0Wait, this is the same equation as before leading to ( m² - 1 ) cos2θ + 2m sin2θ = 0. So that's consistent. So this gives a relationship between θ and m. However, since θ is a parameter that defines the lines, but in the problem, the lines are not given, so m must be such that there exists some θ satisfying this equation. Therefore, we need to find m such that tan2θ = (1 - m²)/(2m). However, since tan2θ can take any real value except when cos2θ = 0, this equation will have a solution as long as (1 - m²)/(2m) is real, which it is. So this doesn't restrict m yet.But perhaps we can combine this with the expressions for r. From equations (2) and (3), we can write:r = | sinθ + m cosθ |,and r = | cosθ - m sinθ |.So, setting them equal:| sinθ + m cosθ | = | cosθ - m sinθ |.Which we already used earlier, leading to the equation involving m and θ. But maybe there's another way to express this.Alternatively, since both expressions equal r, we can square both and set them equal:(sinθ + m cosθ)^2 = (cosθ - m sinθ)^2,Which again gives the same equation as before.Alternatively, perhaps adding them:But since they are absolute values, adding might not help. Alternatively, consider that both are equal to r, so:Either sinθ + m cosθ = cosθ - m sinθ,or sinθ + m cosθ = - (cosθ - m sinθ).Let's consider both cases.Case 1: sinθ + m cosθ = cosθ - m sinθ.Bring all terms to left:sinθ + m cosθ - cosθ + m sinθ = 0,sinθ (1 + m) + cosθ (m - 1) = 0.Divide both sides by cosθ (assuming cosθ ≠ 0):tanθ (1 + m) + (m - 1) = 0,tanθ = (1 - m)/(1 + m).Case 2: sinθ + m cosθ = -cosθ + m sinθ.Bring all terms to left:sinθ + m cosθ + cosθ - m sinθ = 0,sinθ (1 - m) + cosθ (m + 1) = 0.Divide by cosθ:tanθ (1 - m) + (m + 1) = 0,tanθ = -(m + 1)/(1 - m).So in either case, we have tanθ expressed in terms of m. Let's consider both cases.Case 1: tanθ = (1 - m)/(1 + m).Then, using tan2θ = 2 tanθ / (1 - tan²θ).Compute tan2θ:= 2*( (1 - m)/(1 + m) ) / (1 - ( (1 - m)^2 / (1 + m)^2 ) )= 2(1 - m)/(1 + m) / [ ( (1 + m)^2 - (1 - m)^2 ) / (1 + m)^2 )Simplify denominator:(1 + 2m + m²) - (1 - 2m + m²) ) = 4m.So tan2θ = [2(1 - m)/(1 + m)] / [4m / (1 + m)^2 ]= [2(1 - m)/(1 + m)] * [ (1 + m)^2 / 4m ]= [2(1 - m)(1 + m)^2 ] / [4m(1 + m)]= [2(1 - m)(1 + m)] / 4m= [2(1 - m²)] / 4m= (1 - m²)/(2m)Which matches the earlier result. Similarly, for Case 2: tanθ = -(m + 1)/(1 - m).Compute tan2θ:= 2 tanθ / (1 - tan²θ )= 2*(-(m + 1)/(1 - m)) / [1 - ( (m + 1)^2 / (1 - m)^2 ) ]= -2(m + 1)/(1 - m) / [ ( (1 - m)^2 - (m + 1)^2 ) / (1 - m)^2 )Denominator:(1 - 2m + m²) - (m² + 2m + 1) ) = -4m.Thus,tan2θ = [ -2(m + 1)/(1 - m) ] / [ -4m / (1 - m)^2 )= [ -2(m + 1)/(1 - m) ] * [ (1 - m)^2 / (-4m) )= [ 2(m + 1)(1 - m)^2 ] / [4m(1 - m) ]= [ 2(m + 1)(1 - m) ] / 4m= [ -2(m + 1)(m - 1) ] / 4m= [ -2(m² - 1) ] / 4m= (1 - m²)/(2m)Same result. So regardless of case, tan2θ = (1 - m²)/(2m), which matches the previous equation. Therefore, the two cases lead to the same condition.So, the key equation we have is tan2θ = (1 - m²)/(2m). But how does this relate to the other condition involving the radius and circle C?We also know from equation (1) that r = sqrt(9 + m²) - 2. And from the distance to the lines, we have r = | sinθ + m cosθ |. Let me use equation (2):r = | sinθ + m cosθ |.But since we have tanθ from either case, let's express sinθ and cosθ in terms of m. Let's take Case 1: tanθ = (1 - m)/(1 + m). Let me denote t = tanθ = (1 - m)/(1 + m). Then, sinθ = t / sqrt(1 + t²), cosθ = 1 / sqrt(1 + t²). Alternatively, using right triangle with opposite side (1 - m) and adjacent side (1 + m), hypotenuse sqrt( (1 - m)^2 + (1 + m)^2 ) = sqrt(1 - 2m + m² + 1 + 2m + m² ) = sqrt(2 + 2m² ) = sqrt(2(1 + m²)).Therefore, sinθ = (1 - m)/sqrt(2(1 + m²)),cosθ = (1 + m)/sqrt(2(1 + m²)).Plugging into r = | sinθ + m cosθ |:r = | [ (1 - m) + m(1 + m) ] / sqrt(2(1 + m²)) |.Simplify numerator:(1 - m) + m + m² = 1 + m².Therefore,r = | (1 + m²) / sqrt(2(1 + m²)) | = | sqrt(1 + m²) / sqrt(2) | = sqrt( (1 + m²)/2 ).Since m is positive, all terms are positive, so absolute value can be removed:r = sqrt( (1 + m²)/2 ). Equation (4)But from equation (1), we also have r = sqrt(9 + m²) - 2. Therefore:sqrt( (1 + m²)/2 ) = sqrt(9 + m²) - 2.Let me denote sqrt(9 + m²) as S. Then the equation becomes sqrt( (1 + m²)/2 ) = S - 2. But S = sqrt(9 + m²), so S^2 = 9 + m². Therefore, (1 + m²)/2 = (S - 2)^2 = S² - 4S + 4 = (9 + m²) - 4S + 4.Therefore:(1 + m²)/2 = 13 + m² - 4S.Multiply both sides by 2:1 + m² = 26 + 2m² - 8S.Bring all terms to left side:1 + m² - 26 - 2m² + 8S = 0,-25 - m² + 8S = 0,8S = m² + 25,S = (m² + 25)/8.But S = sqrt(9 + m²). Therefore:sqrt(9 + m²) = (m² + 25)/8.Square both sides:9 + m² = (m² + 25)^2 / 64.Multiply both sides by 64:64(9 + m²) = (m² + 25)^2.Compute left side: 576 + 64m².Right side: m^4 + 50m² + 625.Bring all terms to left side:576 + 64m² - m^4 - 50m² - 625 = 0,Simplify:(-m^4) + (64m² -50m²) + (576 - 625) = 0,Which is:-m^4 + 14m² - 49 = 0,Multiply both sides by -1:m^4 -14m² +49 = 0.This is a quadratic in terms of m². Let me set u = m²:u² -14u +49 = 0.Solve for u:u = [14 ± sqrt(196 - 196)] / 2 = [14 ± 0]/2 = 7.Therefore, u = 7, so m² =7. Since m >0, m = sqrt(7).Therefore, m = sqrt(7). Now, compute r using equation (4):r = sqrt( (1 + 7)/2 ) = sqrt(8/2 ) = sqrt(4 ) = 2.Wait, but also from equation (1):r = sqrt(9 + 7) -2 = sqrt(16) -2 =4 -2=2. So consistent. Therefore, radius r is 2.Therefore, the equation of circle M is (x -1)^2 + (y - sqrt(7))^2 = 2^2 =4.But let me check if this makes sense. The center is at (1, sqrt(7)), radius 2. Let's verify the distance to circle C: sqrt( (1 +2)^2 + (sqrt(7) -0)^2 ) = sqrt(9 +7)=sqrt(16)=4, which is 2 + 2, so yes, they are externally tangent. Also, the distance from M(1, sqrt(7)) to lines l1 and l2 should be 2. Let's verify using the earlier expressions.From equation (4), r =2. From Case 1, tanθ=(1 - m)/(1 +m )=(1 -sqrt(7))/(1 +sqrt(7)). Let me compute sinθ and cosθ:From before, sinθ=(1 - m)/sqrt(2(1 +m²))=(1 -sqrt(7))/sqrt(2(1 +7))=(1 -sqrt(7))/sqrt(16)=(1 -sqrt(7))/4.Similarly, cosθ=(1 +m)/sqrt(2(1 +m²))=(1 +sqrt(7))/4.Then, sinθ + m cosθ=(1 -sqrt(7))/4 + sqrt(7)*(1 +sqrt(7))/4.Compute:= [1 - sqrt(7) + sqrt(7) + 7]/4= (1 +7)/4 =8/4=2. Which is equal to r=2. Similarly, cosθ - m sinθ=(1 +sqrt(7))/4 - sqrt(7)*(1 -sqrt(7))/4.Compute:= [1 +sqrt(7) -sqrt(7) +7]/4= (1 +7)/4=8/4=2. Also equal to r=2. So this checks out.Therefore, the equation of circle M is (x -1)^2 + (y - sqrt(7))^2=4.Final AnswerThe equation of circle M is boxed{(x - 1)^2 + left(y - sqrt{7}right)^2 = 4}.</think>Given circle ( C ): ((x+2)^2 + y^2 = 4) with center ((-2, 0)) and radius 2, and two perpendicular lines ( l_1 ) and ( l_2 ) passing through point ( A(2, 0) ). We need to find the equation of circle ( M ) with center ( (1, m) ) (where ( m > 0 )) that is tangent to both circle ( C ) and the lines ( l_1 ) and ( l_2 ).1. Distance between centers of circles ( C ) and ( M ): - The distance between the centers ((-2, 0)) and ((1, m)) is (sqrt{(1 + 2)^2 + m^2} = sqrt{9 + m^2}). - Since the circles are externally tangent, this distance equals the sum of their radii: (sqrt{9 + m^2} = 2 + r), where ( r ) is the radius of circle ( M ). - Thus, ( r = sqrt{9 + m^2} - 2 ).2. Distance from center of circle ( M ) to lines ( l_1 ) and ( l_2 ): - The lines ( l_1 ) and ( l_2 ) are perpendicular and pass through ( A(2, 0) ). The distance from ( (1, m) ) to each line must be equal to ( r ). - Using the relationship between the slopes of the lines and the distances, we derive that the radius ( r ) can also be expressed as ( sqrt{frac{1 + m^2}{2}} ).3. Equating the two expressions for ( r ): - Solving (sqrt{frac{1 + m^2}{2}} = sqrt{9 + m^2} - 2): - Let ( S = sqrt{9 + m^2} ). Then ( sqrt{frac{1 + m^2}{2}} = S - 2 ). - Squaring both sides and solving the resulting equation, we find ( m = sqrt{7} ).4. Radius ( r ) of circle ( M ): - Substituting ( m = sqrt{7} ) into ( r = sqrt{9 + m^2} - 2 ), we get ( r = 2 ).5. Equation of circle ( M ): - The center of circle ( M ) is ( (1, sqrt{7}) ) with radius 2. Thus, the equation is ((x - 1)^2 + (y - sqrt{7})^2 = 4).[boxed{(x - 1)^2 + left(y - sqrt{7}right)^2 = 4}]]

🔑:Let the radius of circle M be r, Since the two tangents l_1 and l_2 from circle M are perpendicular to each other, The distance from the center M(1, m) to point A(2, 0) is sqrt{2}r, Therefore, we have the system of equations: begin{cases} (1-2)^{2}+m^{2}=2r^{2} (1+2)^{2}+m^{2}=(2+r)^{2} end{cases} Solving this, we get r=2 and m= sqrt{7}, Therefore, the equation of circle M is: (x-1)^2+(y-sqrt{7})^2=4 Hence, the answer is: boxed{(x-1)^2+(y-sqrt{7})^2=4} Given the radius of circle M as r, from the problem statement we have the system of equations begin{cases} (1-2)^{2}+m^{2}=2r^{2} (1+2)^{2}+m^{2}=(2+r)^{2} end{cases}, solving this system of equations gives the solution. This problem examines the issue of tangents to a circle, which is a basic question.

🧠:[<think>Okay, so there's this problem about cars moving on a highway. Let me try to understand it step by step. The problem says: A column of cars is moving along the highway at a speed of 80 km/h with a distance of 10 meters between the cars. Then, when they pass a speed limit sign, all cars reduce their speed to 60 km/h. The question is, what will be the distance between the cars in the column after the sign?Hmm. So initially, all cars are moving at 80 km/h with 10 meters between each car. Then they slow down to 60 km/h. We need to find the new distance between them after they've adjusted their speed. First, I need to recall if there's any physics principle that relates speed and distance between vehicles. Maybe something about traffic flow or density? Or maybe it's about the time it takes for each car to pass a certain point, and how the spacing changes when speed changes. Let me think.If all cars are moving at the same speed, the distance between them should depend on their speed and the time interval between each car passing a given point. If they slow down, but the time interval remains the same, the distance between them would decrease. Alternatively, if the spacing adjusts to maintain the same time gap, then the distance would be proportional to speed. Wait, that might be the key here.Let me explain. Suppose each car takes a certain amount of time to pass a point when moving at 80 km/h. The distance between cars is 10 meters. If they slow down to 60 km/h, but maintain the same time interval between cars passing that point, the distance between them would be less because they are moving slower. Alternatively, if they maintain the same distance, the time gap would increase. But in traffic flow, usually, when speed changes, the spacing adjusts to maintain a safe following distance, which is often related to time headway. For example, drivers might maintain a 2-second gap between cars. If that's the case, then when speed decreases, the distance should decrease proportionally.But the problem states that initially, at 80 km/h, the distance between cars is 10 meters. Then after slowing down to 60 km/h, we need to find the new distance. So maybe we need to model this as a change in speed while maintaining the same time gap between cars. Let's work that out.First, let's convert the speeds from km/h to m/s to make the units consistent with the distance in meters.80 km/h is equal to (80 * 1000) meters per hour, which is 80,000 meters per hour. Divided by 3600 seconds in an hour, that's 80,000 / 3600 ≈ 22.222 m/s.Similarly, 60 km/h is 60,000 meters per hour, divided by 3600, which is 60,000 / 3600 ≈ 16.6667 m/s.Okay, so initially, the speed is 22.222 m/s, and the distance between cars is 10 meters. Let's find the time gap between cars. That would be the distance divided by the speed. So 10 meters divided by 22.222 m/s ≈ 0.45 seconds. So roughly a 0.45-second gap between each car. If after slowing down to 16.6667 m/s, they maintain the same time gap, then the new distance between cars would be speed multiplied by time gap. So 16.6667 m/s * 0.45 s ≈ 7.5 meters. Wait, but 0.45 seconds is a very short time gap. In reality, drivers usually maintain a 2-second gap or something like that. But the problem says initially the distance is 10 meters at 80 km/h, so maybe in this problem, we just take the given distance and speed to compute the time gap, then apply it after slowing down. So even if it's not realistic, the problem might be expecting that.Alternatively, maybe the problem is about traffic density. Traffic density is number of cars per unit length. If the speed changes, but the flow remains the same, density would adjust. Wait, flow is speed multiplied by density. But here, all cars reduce speed, so if the flow (vehicles per hour) remains the same, then density would adjust. But in this case, maybe the flow isn't remaining the same? Hmm, I need to be careful here.Wait, the problem says "a column of cars is moving... all cars reduce their speed." So it's the same set of cars, just slowing down. So the number of cars per hour passing a point would decrease because they're moving slower. However, the distance between the cars would change based on their speed if they maintain the same time headway. But the problem is perhaps simpler, assuming that the initial distance is 10 meters at 80 km/h, and when they slow down to 60 km/h, the distance becomes something else. So perhaps proportional to the speed.Alternatively, maybe the time between cars passing a point is the same, so the distance between cars is speed multiplied by time. If they slow down, the distance decreases. But if they all slow down at the same time, maintaining the same spacing, but that's not possible because if the first car slows down, the second car has to slow down too to avoid collision. But in this problem, maybe it's an idealized scenario where all cars reduce speed simultaneously, so the spacing remains the same? Wait, but that contradicts the problem statement. The problem says they reduce speed when passing the sign. So maybe the first car passes the sign and slows down, then the next car passes the sign and slows down, etc. So the distance between them after the sign would depend on how they decelerate. Wait, but the problem doesn't mention deceleration, it just says they reduce speed. So maybe we can assume that they all reduce speed instantaneously when passing the sign. In that case, the spacing would remain the same? But that might not make sense. Because when the first car passes the sign and slows down, the second car is still approaching the sign at 80 km/h. So unless all cars slow down at the same time, which would require them to be at the sign at the same time, which they aren't. So perhaps this is a shock wave problem in traffic flow.Wait, maybe we need to model the propagation of the speed change through the column of cars. When the first car passes the sign and slows down, the next car will have to slow down when it reaches the sign, which is 10 meters behind. So the time between each car slowing down is the time it takes for the next car to reach the sign. So initially, the cars are spaced 10 meters apart at 80 km/h. The time between cars passing the sign is 10 meters divided by 80 km/h. So converting 80 km/h to m/s, as before, 22.222 m/s. So time between cars is 10 / 22.222 ≈ 0.45 seconds. So each car slows down 0.45 seconds after the one in front. Then, after slowing down, each car is moving at 60 km/h (16.666 m/s). So the distance between them would be the speed after slowing down multiplied by the time between cars slowing down. So 16.666 m/s * 0.45 s ≈ 7.5 meters. Alternatively, maybe the distance between the cars is determined by the original time headway. If the time between cars passing a point is 0.45 seconds when moving at 80 km/h, then when they slow down to 60 km/h, the distance between cars would be 60 km/h * 0.45 seconds. But wait, 60 km/h is 16.666 m/s, so 16.666 m/s * 0.45 s ≈ 7.5 meters. So that gives the same result.But let's verify. If the cars are initially spaced 10 meters apart at 80 km/h, the time between cars passing a given point is 10 m / (80 km/h). Convert 80 km/h to m/s: 80 / 3.6 ≈ 22.222 m/s. So time = 10 / 22.222 ≈ 0.45 seconds. If the cars slow down to 60 km/h (16.666 m/s), then the new distance between them would be 16.666 m/s * 0.45 s ≈ 7.5 meters. So that seems to be the answer.Alternatively, maybe we can think about the density of traffic. Traffic density is number of cars per kilometer. Initially, with 10 meters between cars, there are 1000 / 10 = 100 cars per kilometer. So density k1 = 100 cars/km. The speed u1 is 80 km/h. The flow q1 = k1 * u1 = 100 * 80 = 8000 cars/h. After slowing down, the speed u2 = 60 km/h. If the flow remains the same (assuming no cars are added or removed), then the new density k2 = q1 / u2 = 8000 / 60 ≈ 133.333 cars/km. Therefore, the new spacing between cars is 1 / k2 = 1 / 133.333 ≈ 0.0075 km = 7.5 meters. So that also gives 7.5 meters. So both methods give the same answer. Therefore, the answer is 7.5 meters.But wait, let me make sure. The first approach was time headway: original time between cars is 0.45 seconds. After slowing down, distance is speed * time = 16.666 * 0.45 ≈ 7.5 m. The second approach was traffic flow: initial flow is 8000 cars/h, after slowing down, density is 8000 / 60 ≈ 133.333 cars/km, so spacing is 7.5 m. Both methods agree.So, the key idea is that the flow of cars (number per hour) remains constant when they slow down, which causes the density to increase, hence the spacing between cars decreases. Alternatively, maintaining the same time gap between cars leads to reduced spacing when speed decreases. Both perspectives lead to the same result. Therefore, the answer should be 7.5 meters, which is 7.5 m.But let me check again. 80 km/h to 60 km/h is a reduction by a factor of 60/80 = 3/4. So if the original distance was 10 meters, the new distance would be 10 * (60/80) = 7.5 meters. So that's the simplest way. Because if the time between cars remains the same, then the distance is proportional to speed. So the answer is 7.5 meters. Yes, that makes sense. Because if each car takes a certain time to reach the sign after the previous one, and they all slow down to 60 km/h, then the distance between them would be the speed multiplied by that same time. Since time = original distance / original speed, new distance = new speed * time = new speed * (original distance / original speed) = original distance * (new speed / original speed). So 10 m * (60/80) = 7.5 m. That's straightforward.So regardless of the method, the answer is 7.5 meters. Therefore, the distance between the cars after the sign will be 7.5 meters.Final AnswerThe distance between the cars after the sign will be boxed{7.5} meters.</think>Initially, the cars are moving at a speed of 80 km/h with a distance of 10 meters between them. When they reduce their speed to 60 km/h, we need to determine the new distance between them.1. Convert speeds to meters per second: - 80 km/h = (frac{80 times 1000}{3600} approx 22.222 , text{m/s}) - 60 km/h = (frac{60 times 1000}{3600} approx 16.667 , text{m/s})2. Calculate the time gap between cars at the initial speed: - Time gap = (frac{text{Distance}}{text{Speed}} = frac{10 , text{m}}{22.222 , text{m/s}} approx 0.45 , text{seconds})3. Determine the new distance using the time gap and reduced speed: - New distance = (text{Speed} times text{Time gap} = 16.667 , text{m/s} times 0.45 , text{s} approx 7.5 , text{meters})4. Alternative method using proportionality: - The distance between cars is proportional to the speed ratio: - New distance = (10 , text{m} times frac{60}{80} = 7.5 , text{meters})Both methods confirm that the new distance between the cars after reducing their speed is (boxed{7.5}) meters.,<think>Alright, let's tackle this problem. So, there's a column of cars moving at 80 km/h with 10 meters between each car. Then they all reduce their speed to 60 km/h after passing a speed limit sign. We need to find the new distance between the cars after the speed reduction. Hmm, okay.First, I need to visualize the scenario. Imagine a line of cars each moving at 80 km/h, maintaining a 10-meter gap between them. When they see the speed limit sign, they all slow down to 60 km/h. The question is, does the distance between them stay the same, increase, or decrease? Intuitively, if all cars slow down at the same time and maintain the same relative speed, maybe the distance between them would stay the same? But that might not be the case. Wait, maybe it's related to the speed and the time it takes for each car to react? But the problem says all cars reduce their speed when passing the sign. So, maybe they all slow down simultaneously? Hmm.Wait, but in reality, drivers take some time to react, but the problem states that all cars reduce their speed upon passing the sign. So each car slows down as it passes the sign. Therefore, the first car slows down when it passes the sign, then the next car will slow down when it reaches the sign, and so on. So there might be a compression or expansion of the distance between the cars as they slow down.Alternatively, maybe it's a steady-state situation. If all cars are moving at 80 km/h with 10 meters apart, and then after the sign, they all move at 60 km/h. Assuming they maintain the same time gap between each other, the distance should adjust according to the new speed. Wait, traffic flow theory might be relevant here. The distance between cars is related to their speed to maintain the same time headway. Time headway is the time between successive cars passing a given point. If the speed changes but the time headway remains the same, the distance between cars would change proportionally to the speed.So, initially, at 80 km/h, the distance between cars is 10 meters. The time headway can be calculated by dividing the distance by the speed. Let me compute that. First, convert 80 km/h to meters per second. 80 km/h is 80,000 meters per hour, divided by 3600 seconds, which is approximately 22.222 m/s. Then, time headway is distance divided by speed, so 10 meters / 22.222 m/s ≈ 0.45 seconds. So each car is approximately 0.45 seconds behind the one in front.After slowing down to 60 km/h, which is 60,000 meters per hour divided by 3600 seconds, that's approximately 16.666 m/s. If the time headway remains the same (0.45 seconds), then the new distance between cars would be speed multiplied by time headway. So 16.666 m/s * 0.45 s ≈ 7.5 meters. Wait, that would mean the distance decreases. But that contradicts my initial thought that if all cars slow down simultaneously, the distance might stay the same. But according to this calculation, the distance would decrease. But why?Wait, perhaps the key here is that when the cars slow down, the driver's reaction time or the deceleration isn't considered here. The problem states that all cars reduce their speed when passing the sign, which might mean that each car slows down exactly at the point of the sign. So the first car passes the sign and slows down to 60 km/h. The next car, which is 10 meters behind, will pass the sign a bit later, and then slow down. The time between each car passing the sign is the original time headway, which was 0.45 seconds. So, after slowing down, each subsequent car will be 0.45 seconds behind the previous one, but at the slower speed.Therefore, the distance between the cars after the sign would be the new speed multiplied by the original time headway. Which would be 16.666 m/s * 0.45 s ≈ 7.5 meters. Wait, so according to this, the distance decreases. But this seems counterintuitive because if all cars slow down, you might expect the distance to increase if they maintain the same following distance, but since they're going slower, they might close the gap. Hmm.Alternatively, maybe the flow of traffic (vehicles per hour) remains constant? Flow is typically calculated as speed multiplied by density. Density is vehicles per kilometer. So initially, at 80 km/h, with 10 meters between cars, the density is 1000 meters per kilometer divided by 10 meters per car, which is 100 cars per km. Therefore, flow is 80 km/h * 100 cars/km = 8000 cars/h. After slowing down to 60 km/h, if the flow remains the same, then the new density would be flow divided by speed, so 8000 cars/h / 60 km/h ≈ 133.33 cars/km. Therefore, the distance between cars would be 1000 meters / 133.33 cars ≈ 7.5 meters. So same result. Therefore, the distance becomes 7.5 meters. So this seems to confirm the previous calculation.But is flow conserved in this scenario? If all cars reduce speed at the sign, the flow downstream (after the sign) would be 60 km/h * density. But why would the flow remain the same as before? In reality, when there's a bottleneck, flow can decrease. But since all cars are uniformly slowing down, perhaps the flow does remain the same. Wait, maybe not. If the speed reduces but density increases, flow could stay the same. But in traffic flow, the fundamental relationship is flow = speed * density. So if speed decreases and density increases appropriately, flow can stay the same. But in this case, maybe the flow is determined by the incoming vehicles. Since the vehicles are arriving at the point of the speed limit sign at the original flow rate (8000 cars/h), then after the sign, they continue at a lower speed but higher density, so flow remains the same. Therefore, the density must adjust to maintain the flow. Hence, 8000 cars/h = 60 km/h * density. Therefore, density ≈ 133.33 cars/km, so spacing is 7.5 meters. Therefore, the answer is 7.5 meters.Alternatively, maybe the problem is more straightforward. If the cars were spaced 10 meters apart at 80 km/h, then when they slow down to 60 km/h, the distance between them should adjust so that the time between cars passing a given point remains the same. That time is 10 meters / (80 km/h). Convert 80 km/h to m/s: 80 / 3.6 ≈ 22.222 m/s. Time between cars is 10 / 22.222 ≈ 0.45 seconds. At 60 km/h, which is 16.666 m/s, the distance required to maintain 0.45 seconds is 16.666 * 0.45 ≈ 7.5 meters. Therefore, same answer.So both approaches give 7.5 meters. Therefore, the answer is 7.5 meters. But wait, let me double-check.Original speed: 80 km/h. Distance between cars: 10 m. Time between cars passing a point: 10 m / (80,000 m / 3600 s) = 10 / (22.222) ≈ 0.45 s. After slowing to 60 km/h, which is 16.666 m/s, the distance between cars would be 16.666 m/s * 0.45 s ≈ 7.5 m. So yes, this seems correct.Alternatively, if drivers maintain the same following distance, but since speed is lower, the time headway would increase. But the problem states that all cars reduce speed when passing the sign. So, each car slows down at the sign, so the distance between cars after the sign would depend on how they decelerate. Wait, maybe I need to consider the deceleration as well? But the problem doesn't mention anything about how quickly they slow down, just that they reduce speed to 60 km/h after passing the sign. Hmm. If they brake abruptly to 60 km/h at the sign, then the distance might stay the same? But in reality, if the first car brakes, the following car would have to brake as well, potentially leading to a compression wave. But since all cars brake at the same location (the sign), maybe they all just continue with the new speed, maintaining the original time headway. But since they slow down, the distance would decrease. So, perhaps the answer is indeed 7.5 meters.Alternatively, if the cars are spaced 10 meters apart at 80 km/h, and then they slow down to 60 km/h, assuming they keep the same safety distance, which is usually time-based, like 2 seconds, then the distance would be speed * time. But the initial distance is 10 meters, which corresponds to a time of 10/(80/3.6)= 10/22.222≈0.45 seconds. If drivers maintain the same time gap, then at 60 km/h, the distance would be 60/3.6 * 0.45 ≈ 16.666 * 0.45 ≈7.5 meters. So same result. So perhaps the answer is 7.5 meters.Therefore, after analyzing through different approaches—traffic flow, time headway, and driver behavior—it seems consistent that the distance reduces to 7.5 meters. Therefore, the answer should be 7.5 meters. But let me check the problem statement again."A column of cars is moving along the highway at a speed of 80 km/h with a distance of 10 m between the cars. Passing a speed limit sign, all cars reduce their speed to 60 km/h. What will be the distance between the cars in the column after the sign?"All cars reduce speed upon passing the sign. So the lead car passes the sign, slows down. The next car is 10 meters behind at 80 km/h, then it passes the sign and slows down. The time between the first and second car passing the sign is 10 meters / (80 km/h). Then, after slowing down, the second car is traveling at 60 km/h. The distance between them after the sign would be the speed difference multiplied by the time? Wait, if both cars are now going 60 km/h, then once both have passed the sign, their speeds are the same, so the distance between them should remain the same as when they passed the sign. But when the first car passes the sign, it slows to 60. The second car is 10 meters behind at 80 km/h. The time it takes for the second car to reach the sign is (10 meters) / (80 km/h converted to m/s) = 10 / 22.222 ≈ 0.45 seconds. During that time, the first car is moving at 60 km/h, which is 16.666 m/s. So in 0.45 seconds, the first car would have moved 16.666 * 0.45 ≈7.5 meters from the sign. Therefore, when the second car passes the sign, the first car is 7.5 meters ahead. Then, both cars continue at 60 km/h, so the distance remains 7.5 meters. Hence, the new distance is 7.5 meters. Therefore, the answer is 7.5 meters.Yes, this makes sense. The key is that the time it takes for each subsequent car to reach the sign after the previous one is 0.45 seconds. During that time, the previous car has already slowed down and moved 7.5 meters away. Therefore, the new spacing is 7.5 meters. Therefore, the answer is 7.5 meters. Converting that to a boxed number, it would be boxed{7.5} meters.Final AnswerThe distance between the cars after the sign will be boxed{7.5} meters.</think>The problem involves a column of cars moving at 80 km/h with a distance of 10 meters between each car. Upon passing a speed limit sign, all cars reduce their speed to 60 km/h. We need to determine the new distance between the cars after the speed reduction.1. Convert speeds to meters per second: - Initial speed: ( 80 , text{km/h} = frac{80 times 1000}{3600} approx 22.222 , text{m/s} ) - New speed: ( 60 , text{km/h} = frac{60 times 1000}{3600} approx 16.666 , text{m/s} )2. Calculate the time headway at the initial speed: - Time headway is the time between successive cars passing a given point. - Time headway: ( frac{10 , text{meters}}{22.222 , text{m/s}} approx 0.45 , text{seconds} )3. Determine the new distance at the reduced speed: - The time headway remains the same after the speed reduction. - New distance: ( 16.666 , text{m/s} times 0.45 , text{seconds} approx 7.5 , text{meters} )4. Verification using traffic flow theory: - Initial flow: ( 80 , text{km/h} times 100 , text{cars/km} = 8000 , text{cars/h} ) - New density: ( frac{8000 , text{cars/h}}{60 , text{km/h}} approx 133.33 , text{cars/km} ) - New distance: ( frac{1000 , text{meters}}{133.33 , text{cars}} approx 7.5 , text{meters} )Both methods confirm that the new distance between the cars after the speed reduction is ( 7.5 , text{meters} ).[boxed{7.5}]]

🔑:Answer: 7.5 m. Solution. Let the speeds be v_{1}=80 km/h, v_{2}=60 km/h, and the distance a=10 m =0.01 km. The second car passes the sign later than the first by frac{a}{v_{1}} (hours). During this time, the first car will travel a distance of v_{2} cdot frac{a}{v_{1}}=7.5 text{ m}, with this distance the cars will continue to move further.

🧠:[<think>Okay, so I need to figure out whether this fraction (frac{a^2 + b^2 - c^2 + 2ab}{a^2 + c^2 - b^2 + 2ac}) is reducible or not, and if it is, which of the options it matches. The answer choices are from A to E, with E being reducible to (frac{a + b - c}{a - b + c}). Let me start by looking at the numerator and the denominator separately. Maybe I can factor them or simplify them somehow.First, let's look at the numerator: (a^2 + b^2 - c^2 + 2ab). Hmm. The first three terms, (a^2 + b^2 - c^2), remind me of the cosine law in trigonometry, but I don't know if that's relevant here. The extra term is (2ab). Let me rearrange the terms: (a^2 + 2ab + b^2 - c^2). Oh, wait! (a^2 + 2ab + b^2) is a perfect square. That factors to ((a + b)^2). So the numerator becomes ((a + b)^2 - c^2). That's a difference of squares, which factors further into ((a + b - c)(a + b + c)). Nice, so numerator factors to ((a + b - c)(a + b + c)).Now, let's check the denominator: (a^2 + c^2 - b^2 + 2ac). Similarly, rearrange the terms: (a^2 + 2ac + c^2 - b^2). Again, (a^2 + 2ac + c^2) is a perfect square, which is ((a + c)^2). So the denominator becomes ((a + c)^2 - b^2), another difference of squares. That factors into ((a + c - b)(a + c + b)).So now the fraction is (frac{(a + b - c)(a + b + c)}{(a + c - b)(a + c + b)}). Let's write that out:[frac{(a + b - c)(a + b + c)}{(a + c - b)(a + c + b)}]Looking at this, both the numerator and the denominator have a common factor of ((a + b + c)). Wait, the denominator has ((a + c + b)), which is the same as ((a + b + c)). So that term cancels out from both the numerator and the denominator. That leaves us with (frac{a + b - c}{a + c - b}).But let me check: the denominator after factoring is ((a + c - b)(a + c + b)). So the denominator factors are ((a + c - b)) and ((a + c + b)). The numerator factors are ((a + b - c)) and ((a + b + c)). So when we cancel out the ((a + b + c)) terms, we're left with (frac{a + b - c}{a + c - b}). Now, let's compare this simplified form to the answer choices. The original problem had answer choice D as (frac{a - b + c}{a + b - c}) and E as (frac{a + b - c}{a - b + c}). Wait, the simplified fraction here is (frac{a + b - c}{a + c - b}). Let me rearrange the denominator's terms: (a + c - b) can be written as (a - b + c). So the simplified fraction is (frac{a + b - c}{a - b + c}), which is exactly option E. Let me verify the steps again to make sure I didn't make a mistake. First, factoring the numerator:Original numerator: (a^2 + b^2 - c^2 + 2ab). Rearranged as (a^2 + 2ab + b^2 - c^2), which becomes ((a + b)^2 - c^2), then factors to ((a + b - c)(a + b + c)). That seems correct.Denominator: (a^2 + c^2 - b^2 + 2ac). Rearranged as (a^2 + 2ac + c^2 - b^2), which is ((a + c)^2 - b^2), factors to ((a + c - b)(a + c + b)). Correct again.Canceling out the common ((a + b + c)) term gives (frac{a + b - c}{a + c - b}). Then rewriting the denominator as (a - b + c) gives the fraction (frac{a + b - c}{a - b + c}), which matches option E. So the correct answer should be E.But just to be thorough, let me check if there are any other factors or possible simplifications I might have missed. For instance, could (a + b - c) and (a - b + c) have a common factor? If they did, then perhaps the fraction could be simplified further, but in general, without specific relationships between a, b, and c, these are linear terms and are irreducible. So unless there's a specific value of a, b, c that allows further simplification, the fraction reduces to option E.Another check: Let's plug in some numbers to test. Suppose a=2, b=1, c=1.Original numerator: 2² + 1² -1² + 2*2*1 = 4 +1 -1 +4 = 8Original denominator: 2² +1² -1² + 2*2*1 = 4 +1 -1 +4 =8Wait, plugging in a=2, b=1, c=1 gives 8/8=1. Let's see what option E would give:(a + b - c)/(a - b + c) = (2 +1 -1)/(2 -1 +1) = (2)/(2)=1. That matches. If I choose different numbers, say a=3, b=2, c=1.Original numerator: 9 +4 -1 +12=24Original denominator:9 +1 -4 +12=18So fraction is 24/18=4/3.Option E: (3+2-1)/(3-2+1)=4/2=2. Wait, that doesn't match 4/3. Wait, that's a problem. Did I do the calculations right?Wait, let's recheck. For a=3, b=2, c=1:Numerator: a² + b² - c² + 2ab = 9 +4 -1 +12=24Denominator: a² + c² - b² + 2ac=9 +1 -4 +6=12Wait, 9 +1 -4 +6=12? Wait, 9 +1=10, 10-4=6, 6+6=12. So denominator is 12. So the fraction is 24/12=2. Option E gives (3 +2 -1)/(3 -2 +1)=4/2=2. That does match. Wait, so my calculation earlier was wrong. Wait, in denominator: 2ac for a=3, c=1 is 2*3*1=6. So yes, denominator is 9 +1 -4 +6=12. So fraction is 24/12=2, and option E gives 4/2=2. Correct.Wait, in my previous test case, a=2, b=1, c=1. Numerator: 4 +1 -1 +4=8. Denominator:4 +1 -1 +4=8. So 8/8=1. Option E: (2+1-1)/(2-1+1)=2/2=1. Correct. Another test case: a=1, b=1, c=1.Numerator:1 +1 -1 +2=3. Denominator:1 +1 -1 +2=3. So 3/3=1. Option E: (1+1-1)/(1-1+1)=1/1=1. Correct.Another test case where a=4, b=3, c=2.Numerator:16 +9 -4 +24=45. Denominator:16 +4 -9 +16=27. So 45/27=5/3. Option E: (4+3-2)/(4-3+2)=5/3. Correct.So in these test cases, it works out. Therefore, the simplified fraction is indeed option E. So the answer must be E.But just to be thorough, let's check another case where maybe a + b - c is zero. Let's see. Suppose a=1, b=1, c=3. Then numerator becomes 1 +1 -9 +2= -5. Denominator:1 +9 -1 +6=15. So fraction is -5/15= -1/3. Option E: (1 +1 -3)/(1 -1 +3)= (-1)/3= -1/3. Correct. So even when the result is negative, it still works.Alternatively, if a=1, b=2, c=3. Numerator:1 +4 -9 +4=0. Denominator:1 +9 -4 +6=12. So fraction is 0/12=0. Option E: (1 +2 -3)/(1 -2 +3)=0/2=0. Correct. So if numerator is zero, the fraction is zero, which matches.Therefore, after multiple checks, it seems that the fraction simplifies to option E.Final Answerboxed{E}</think>To determine whether the fraction (frac{a^2 + b^2 - c^2 + 2ab}{a^2 + c^2 - b^2 + 2ac}) is reducible and to which form it reduces, we start by examining the numerator and the denominator separately.First, consider the numerator (a^2 + b^2 - c^2 + 2ab). Rearranging the terms, we get:[a^2 + 2ab + b^2 - c^2 = (a + b)^2 - c^2]This is a difference of squares, which factors as:[(a + b - c)(a + b + c)]Next, consider the denominator (a^2 + c^2 - b^2 + 2ac). Rearranging the terms, we get:[a^2 + 2ac + c^2 - b^2 = (a + c)^2 - b^2]This is also a difference of squares, which factors as:[(a + c - b)(a + c + b)]Now, the fraction becomes:[frac{(a + b - c)(a + b + c)}{(a + c - b)(a + c + b)}]We can cancel the common factor ((a + b + c)) from both the numerator and the denominator, leaving us with:[frac{a + b - c}{a + c - b}]Rewriting the denominator (a + c - b) as (a - b + c), we get:[frac{a + b - c}{a - b + c}]This matches option E. Therefore, the correct answer is:[boxed{E}],<think>Okay, so I need to figure out if the fraction (a² + b² - c² + 2ab)/(a² + c² - b² + 2ac) is reducible to one of the options given. Let me start by looking at the numerator and the denominator separately to see if they can be simplified or factored.First, let's take the numerator: a² + b² - c² + 2ab. Hmm, the first three terms remind me of the formula for the cosine law, but I don't know if that's relevant here. Let me see if I can rearrange the terms to make it more familiar. If I group a² + 2ab + b², that's a perfect square, right? So, a² + 2ab + b² = (a + b)². Then, subtracting c² gives me (a + b)² - c². Oh, that's a difference of squares! So, the numerator can be written as (a + b - c)(a + b + c). Let me check that:(a + b - c)(a + b + c) = (a + b)² - c² = a² + 2ab + b² - c². Yes, that's exactly the numerator. Good.Now, the denominator is a² + c² - b² + 2ac. Let me try the same approach. Group a² + 2ac + c², which is (a + c)², then subtract b². So, (a + c)² - b². That's another difference of squares. Therefore, the denominator is (a + c - b)(a + c + b). Checking:(a + c - b)(a + c + b) = (a + c)² - b² = a² + 2ac + c² - b². Perfect, matches the denominator.So now, the fraction becomes:Numerator: (a + b - c)(a + b + c)Denominator: (a + c - b)(a + c + b)Wait, notice that (a + b + c) is the same as (a + c + b), right? So those terms are identical. So, they can be canceled out from the numerator and the denominator. Let me write that:[(a + b - c)(a + b + c)] / [(a + c - b)(a + c + b)] = [(a + b - c)/(a + c - b)] * [(a + b + c)/(a + c + b)] = (a + b - c)/(a + c - b) * 1.Because (a + b + c) is the same as (a + c + b), so that term cancels out to 1. Therefore, the entire fraction simplifies to (a + b - c)/(a + c - b).But let me check the answer options. Option E is reducible to (a + b - c)/(a - b + c). Wait, my simplified result is (a + b - c)/(a + c - b). Let's rearrange the denominator:a + c - b = a - b + c. So yes, that's the same as (a - b + c). Therefore, the simplified fraction is (a + b - c)/(a - b + c), which is option E. Let me confirm:Original fraction after factoring and canceling:(a + b - c)/(a + c - b) = (a + b - c)/(a - b + c). Yes, because a + c - b is the same as a - b + c. So, that's exactly the expression in option E. Therefore, the answer should be E.But wait, let me check if there's any possibility that I made a mistake in factoring. Let me redo the factoring steps quickly.Numerator: a² + b² - c² + 2ab. Group a² + 2ab + b² = (a + b)². Then subtract c²: (a + b)² - c² = (a + b - c)(a + b + c). That's correct.Denominator: a² + c² - b² + 2ac. Group a² + 2ac + c² = (a + c)². Then subtract b²: (a + c)² - b² = (a + c - b)(a + c + b). Correct. Then, (a + b + c) cancels with (a + c + b). So, we have (a + b - c)/(a + c - b) = (a + b - c)/(a - b + c). Yes, because addition is commutative, so a + c - b is the same as a - b + c.Therefore, the simplified fraction is (a + b - c)/(a - b + c), which is option E. Let me look at the options again:A) irreducible – No, we reduced it.B) reducible to negative 1 – No, unless numerator and denominator are negatives of each other. Let me check: if (a + b - c) = -(a - b + c), then a + b - c = -a + b - c → a = -a → 2a = 0 → a = 0. But a is a variable, so unless a is zero, which isn't specified. So generally, they are not negatives. So B is incorrect.C) reducible to a polynomial of three terms – Hmm, after simplifying, it's still a rational expression, not a polynomial. So C is incorrect.D) reducible to (a - b + c)/(a + b - c) – That would be the reciprocal of E. But our result is (a + b - c)/(a - b + c), so D is the reciprocal. Unless there was a negative sign or something. Let me check: If we had made a mistake in the sign. Let me see:Wait, if you look at the denominator factors: (a + c - b) is the same as (a - b + c), correct. So the denominator is (a - b + c), numerator is (a + b - c). So the simplified fraction is (a + b - c)/(a - b + c). So that's exactly E. Therefore, D is the reciprocal, which is not the case here. So D is incorrect.E) is correct.Therefore, the answer is E.Final Answerboxed{E}</think>To determine if the fraction (frac{a^2 + b^2 - c^2 + 2ab}{a^2 + c^2 - b^2 + 2ac}) can be simplified, we start by examining the numerator and the denominator separately.First, consider the numerator (a^2 + b^2 - c^2 + 2ab):- We recognize that (a^2 + 2ab + b^2) forms a perfect square, ((a + b)^2).- Subtracting (c^2) from this gives us a difference of squares: ((a + b)^2 - c^2 = (a + b - c)(a + b + c)).Next, consider the denominator (a^2 + c^2 - b^2 + 2ac):- Similarly, (a^2 + 2ac + c^2) forms a perfect square, ((a + c)^2).- Subtracting (b^2) from this gives another difference of squares: ((a + c)^2 - b^2 = (a + c - b)(a + c + b)).Substituting these factored forms back into the original fraction, we get:[frac{(a + b - c)(a + b + c)}{(a + c - b)(a + c + b)}]Notice that ((a + b + c)) is the same as ((a + c + b)), so these terms cancel out. This leaves us with:[frac{a + b - c}{a + c - b}]Rearranging the terms in the denominator, we recognize that (a + c - b) is equivalent to (a - b + c). Thus, the simplified fraction is:[frac{a + b - c}{a - b + c}]This matches option (E). Therefore, the answer is:[boxed{E}]]

🔑:Use the commutative property to get[frac{a^2+2ab+b^2-c^2}{a^2+2ac+c^2-b^2}]Factor perfect square trinomials to get[frac{(a+b)^2-c^2}{(a+c)^2-b^2}]Factor difference of squares to get[frac{(a+b+c)(a+b-c)}{(a+b+c)(a-b+c)}]Cancel out like terms (with suitable restrictions of a, b, c) to get[frac{a+b-c}{a-b+c}]The answer is boxed{textbf{(E)}}.