🧠:[<think>Okay, let me try to solve this problem step by step. The question is about finding the equation of the locus of a moving point M(x, y) such that the ratio of its distance to the point F(4, 0) to its distance to the line x = 3 is 2. The answer choices are given, so after deriving the equation, I can check which one matches.First, I need to recall that this type of problem is related to conic sections, specifically the definition of a hyperbola or an ellipse based on the ratio of distances. Since the ratio given is 2, which is greater than 1, this should be a hyperbola. For a hyperbola, the ratio (eccentricity) e is greater than 1, and the definition is the set of points where the distance to a focus divided by the distance to the directrix is equal to e.So, let's denote the moving point as M(x, y). The distance from M to the point F(4, 0) can be calculated using the distance formula. The distance from M to the line x = 3 is the horizontal distance, which is |x - 3| since the line is vertical.The ratio given is 2, so according to the problem, distance to F divided by distance to the line x = 3 equals 2. Let's write that equation:distance(M, F) / distance(M, line x=3) = 2So, substituting the formulas:√[(x - 4)^2 + (y - 0)^2] / |x - 3| = 2Multiplying both sides by |x - 3| to eliminate the denominator:√[(x - 4)^2 + y^2] = 2|x - 3|Now, let's square both sides to remove the square root:[(x - 4)^2 + y^2] = (2|x - 3|)^2Since squaring removes the absolute value, the right-hand side becomes 4(x - 3)^2.So, expanding both sides:Left-hand side (LHS): (x^2 - 8x + 16) + y^2Right-hand side (RHS): 4(x^2 - 6x + 9)So, expanding RHS:4x^2 - 24x + 36Now, set up the equation:x^2 - 8x + 16 + y^2 = 4x^2 - 24x + 36Bring all terms to the left-hand side:x^2 - 8x + 16 + y^2 - 4x^2 + 24x - 36 = 0Combine like terms:(1x^2 - 4x^2) + (-8x + 24x) + (16 - 36) + y^2 = 0(-3x^2) + (16x) + (-20) + y^2 = 0Rearranging terms:-3x^2 + y^2 + 16x - 20 = 0Multiply both sides by -1 to make the coefficients more positive:3x^2 - y^2 - 16x + 20 = 0Looking at the options given, option C is "3x² - y² -16x +20=0", which matches this equation. However, let me check if I made any mistakes in the algebra.Let me go through the steps again.Starting with the ratio:√[(x - 4)^2 + y^2] / |x - 3| = 2Multiplying both sides by |x - 3|:√[(x - 4)^2 + y^2] = 2|x - 3|Squaring both sides:(x - 4)^2 + y^2 = 4(x - 3)^2Expanding both sides:Left: x² - 8x + 16 + y²Right: 4(x² - 6x + 9) = 4x² - 24x + 36Subtracting RHS from LHS:x² -8x +16 + y² -4x² +24x -36 = -3x² +16x -20 + y² = 0Which is the same as:3x² - y² -16x +20 = 0 when multiplied by -1.Yes, that seems correct. So, the equation is 3x² - y² -16x +20=0, which is option C.But wait, let me check the answer choices again. Option C is exactly that: "3x² - y² -16x +20=0". The other options are in standard hyperbola form. Let me see if perhaps this equation can be converted to standard form to check if it matches options A or B.Starting with equation C:3x² - y² -16x +20 = 0Let's complete the square for the x terms.Group the x terms and the constant:3x² -16x - y² +20 =0Factor out the coefficient of x² from the x terms:3(x² - (16/3)x) - y² +20 =0To complete the square inside the parentheses:Take half of (16/3), which is 8/3, then square it: (8/3)² = 64/9So, add and subtract 64/9 inside the parentheses:3[(x² - (16/3)x + 64/9 - 64/9) ] - y² +20 =0Which becomes:3[(x - 8/3)² - 64/9] - y² +20 =0Multiply through by 3:3(x - 8/3)² - 64/3 - y² +20 =0Combine constants:Convert 20 to 60/3 to have a common denominator:-64/3 + 60/3 = -4/3So:3(x - 8/3)² - y² -4/3 =0Multiply both sides by 3 to eliminate denominators:9(x - 8/3)² - 3y² -4 =0But maybe this isn't helpful. Alternatively, let's rearrange the equation to standard hyperbola form.Starting again from 3x² - y² -16x +20 =0Bring constants to the right:3x² -16x - y² = -20Complete the square for x:3x² -16x = 3(x² - (16/3)x)Half of 16/3 is 8/3, square is 64/9.So:3[ x² - (16/3)x + 64/9 - 64/9 ] = 3[ (x - 8/3)^2 - 64/9 ]= 3(x - 8/3)^2 - (3)(64/9) = 3(x - 8/3)^2 - 64/3So substituting back into equation:3(x - 8/3)^2 - 64/3 - y² = -20Bring -64/3 to the right:3(x - 8/3)^2 - y² = -20 + 64/3Convert -20 to -60/3:= -60/3 +64/3 = 4/3So:3(x - 8/3)^2 - y² = 4/3Divide both sides by 4/3 to get 1 on the right:[3(x - 8/3)^2]/(4/3) - y²/(4/3) =1Simplify:Multiply numerator and denominator:3/(4/3) = 9/4, so the first term becomes (9/4)(x - 8/3)^2Similarly, the second term is - (y²)/(4/3) = - (3/4)y²Thus, the equation is:(9/4)(x - 8/3)^2 - (3/4)y² =1Multiply both sides by 4 to eliminate denominators:9(x - 8/3)^2 - 3y² =4But this still doesn't look like the standard hyperbola form because the coefficients aren't 1. Let me see:Divide both sides by 4:[9(x - 8/3)^2]/4 - [3y²]/4 =1But maybe this is not helpful. Alternatively, perhaps my approach is complicating things. Let me check the answer choices again.Option C is 3x² - y² -16x +20=0, which is the equation we derived. Option B is (x²)/4 - (y²)/12=1, and Option A is (x²)/12 - (y²)/4=1.If we write the standard form of the hyperbola, after completing the square, maybe we can see if it matches A or B. Let's try again.Starting from 3x² - y² -16x +20=0We can rearrange terms:3x² -16x - y² = -20Complete the square for the x terms:Factor out 3 from the x terms:3(x² - (16/3)x) - y² = -20Complete the square inside the parentheses:x² - (16/3)x + (8/3)^2 - (8/3)^2 = (x - 8/3)^2 - 64/9So:3[(x - 8/3)^2 - 64/9] - y² = -20Multiply through:3(x - 8/3)^2 - 64/3 - y² = -20Bring constants to the right:3(x - 8/3)^2 - y² = -20 + 64/3Convert -20 to -60/3:= (-60/3 +64/3) = 4/3So:3(x - 8/3)^2 - y² = 4/3Divide both sides by 4/3 to normalize:[3(x - 8/3)^2]/(4/3) - y²/(4/3) =1Simplify:3/(4/3) = 9/4, so:(9/4)(x - 8/3)^2 - (3/4)y² =1Which can be written as:[(x - 8/3)^2]/(4/9) - [y²]/(4/3) =1But 4/9 is (2/3)^2 and 4/3 is (2/√3)^2. So the standard form is:[(x - 8/3)^2]/( (2/3)^2 ) - [y²]/( (2/√3)^2 ) =1This is a hyperbola centered at (8/3, 0), with a horizontal transverse axis. However, none of the answer choices A or B are in this form; they are centered at the origin. The given options C and D are not in standard form but are the expanded forms. Since we derived option C as the correct equation before completing the square, and the problem doesn't specify that the equation must be in standard form, just the equation of the locus, which can be in general form, then option C is the answer.But let me check if perhaps there was a miscalculation when squaring both sides. Let me verify that step again.Original equation after squaring:(x - 4)^2 + y^2 = 4(x - 3)^2Expanding left side: x² -8x +16 + y²Expanding right side: 4(x² -6x +9) = 4x² -24x +36Moving all terms to left: x² -8x +16 + y² -4x² +24x -36 =0Combine like terms: (1 -4)x² + (-8x +24x) + (16 -36) + y² = -3x² +16x -20 + y² =0Which leads to 3x² - y² -16x +20 =0 when multiplied by -1. Correct.Therefore, the equation is 3x² - y² -16x +20 =0, which is option C. So the answer should be C.But just to confirm, let me check if option C is indeed a hyperbola. Let's see:3x² - y² -16x +20=0If we complete the square as we did earlier, it's a hyperbola. The original ratio being 2 (>1) also indicates it's a hyperbola, so the equation must represent a hyperbola, and among the options, both A and B are hyperbolas in standard form, but centered at the origin. However, our derived equation is not centered at the origin, so it's not A or B, which are centered at (0,0). Therefore, the correct answer must be C.Final Answerboxed{C}</think>Given a moving point ( M(x, y) ) such that the ratio of its distance to the point ( F(4, 0) ) to its distance to the line ( x = 3 ) is 2, we need to find the equation of the locus of ( M ).1. Set up the ratio equation: [ frac{sqrt{(x - 4)^2 + y^2}}{|x - 3|} = 2 ]2. Multiply both sides by ( |x - 3| ): [ sqrt{(x - 4)^2 + y^2} = 2|x - 3| ]3. Square both sides: [ (x - 4)^2 + y^2 = 4(x - 3)^2 ]4. Expand both sides: [ x^2 - 8x + 16 + y^2 = 4(x^2 - 6x + 9) ]5. Simplify the right-hand side: [ x^2 - 8x + 16 + y^2 = 4x^2 - 24x + 36 ]6. Bring all terms to the left-hand side: [ x^2 - 8x + 16 + y^2 - 4x^2 + 24x - 36 = 0 ]7. Combine like terms: [ -3x^2 + 16x - 20 + y^2 = 0 ]8. Multiply by -1 to simplify: [ 3x^2 - y^2 - 16x + 20 = 0 ]The resulting equation matches option (C). Therefore, the equation of the locus of ( M(x, y) ) is:[boxed{C}],<think>Okay, so I need to find the equation of the locus of a point M(x, y) such that the ratio of its distance to the point F(4, 0) to its distance to the line x = 3 is 2. The answer choices are given, so maybe I can derive the equation step by step and see which one matches.First, let's recall the definition of a conic section. A conic is the set of all points where the ratio of the distance to a fixed point (focus) and the distance to a fixed line (directrix) is a constant called the eccentricity. If the eccentricity is greater than 1, it's a hyperbola; equal to 1, it's a parabola; less than 1, it's an ellipse. Here, the ratio is 2, which is greater than 1, so this should be a hyperbola. So the answer is likely one of the hyperbola equations, which are options A, B, C, D. Wait, options C and D don't look like standard hyperbola forms, but A and B are standard. Let's check.Let me start by setting up the equation based on the given ratio.The distance from M(x, y) to F(4, 0) is √[(x - 4)² + (y - 0)²] = √[(x - 4)² + y²].The distance from M(x, y) to the line x = 3. The formula for the distance from a point (x, y) to the vertical line x = a is |x - a|. So here, it's |x - 3|.According to the problem, the ratio of these distances is 2. So:√[(x - 4)² + y²] / |x - 3| = 2.Multiply both sides by |x - 3| to get:√[(x - 4)² + y²] = 2|x - 3|.Now, square both sides to eliminate the square root:[(x - 4)² + y²] = 4(x - 3)².Let me expand both sides.Left side: (x² - 8x + 16) + y².Right side: 4(x² - 6x + 9) = 4x² - 24x + 36.Now, set up the equation:x² - 8x + 16 + y² = 4x² - 24x + 36.Bring all terms to the left side:x² - 8x + 16 + y² - 4x² + 24x - 36 = 0.Combine like terms:(1x² - 4x²) + (-8x + 24x) + (16 - 36) + y² = 0.That's (-3x²) + (16x) + (-20) + y² = 0.So rearranged:-3x² + y² + 16x - 20 = 0.Multiply through by -1 to make the coefficients positive:3x² - y² - 16x + 20 = 0.Hmm, looking at the answer choices, option C is "3x² - y² -16x +20=0", which is exactly what I got here. But wait, let me check my steps again to make sure.Starting from the ratio:√[(x - 4)^2 + y^2] / |x - 3| = 2.Multiplying both sides by |x - 3| gives √[(x -4)^2 + y^2] = 2|x -3|.Squaring both sides: (x -4)^2 + y^2 = 4(x -3)^2.Expanding left: x² -8x +16 + y².Right: 4(x² -6x +9) =4x² -24x +36.Subtracting right side from left: x² -8x +16 + y² -4x² +24x -36 =0.Combine terms: -3x² +16x -20 + y² =0. Rearranged: 3x² - y² -16x +20=0. Yes, that's correct. So equation C.But let's check the answer options. The options are:A. x²/12 - y²/4 =1B. x²/4 - y²/12=1C. 3x² - y² -16x +20=0D. 3y² -x² -16y +20=0So the equation I derived is C, which is not in standard form. Let me see if maybe it can be rewritten into standard form, which might correspond to A or B. Let's try.Starting with 3x² - y² -16x +20=0.Let me complete the square for the x-terms.3x² -16x - y² +20=0.Factor out 3 from the x terms:3(x² - (16/3)x) - y² +20=0.To complete the square for x² - (16/3)x:Take half of 16/3, which is 8/3, square it: (8/3)^2 =64/9.So add and subtract 64/9 inside the parentheses:3[ x² - (16/3)x + 64/9 -64/9 ] - y² +20=0Which is 3[ (x - 8/3)^2 -64/9 ] - y² +20=0Multiply through:3(x -8/3)^2 -3*(64/9) - y² +20=0Simplify:3(x -8/3)^2 - (64/3) - y² +20=0Convert 20 to thirds: 20=60/3So:3(x -8/3)^2 -64/3 -60/3 - y²=0Combine constants:3(x -8/3)^2 - y² -124/3=0Bring constants to the other side:3(x -8/3)^2 - y² =124/3Divide both sides by 124/3 to get 1 on the right:[3(x -8/3)^2]/(124/3) - y²/(124/3) =1Simplify:[ (x -8/3)^2 ]/(124/9) - [ y² ]/(124/3) =1Which is:(x -8/3)^2 / (124/9) - y²/(124/3) =1But 124/9 simplifies to approximately 13.78, and 124/3 is about 41.33. However, these denominators can be expressed as:124/9 = (4*31)/9, and 124/3 = (4*31)/3. Not sure if this reduces. Maybe check if 124 is divisible by something. 124 divided by 4 is 31, so:124/9 = 4*31/9, 124/3=4*31/3. So unless there is a common factor, the equation doesn't simplify to the options A or B, which have denominators 12 and 4 or 4 and 12. So maybe the answer is C. Let's check the options again.Alternatively, perhaps I made a mistake in completing the square. Let me try again.Original equation after moving all terms to left: 3x² - y² -16x +20=0.Let me rearrange terms:3x² -16x - y² +20=0.Factor 3 out of the x terms:3(x² - (16/3)x) - y² +20=0.Complete the square for x² - (16/3)x:Take half of 16/3, which is 8/3, square is 64/9.So:3[(x² - (16/3)x + 64/9) - 64/9] - y² +20=0Which is:3[(x - 8/3)^2 - 64/9] - y² +20=0Multiply through:3(x -8/3)^2 - (3)(64/9) - y² +20=0Compute 3*(64/9) = 64/3.So:3(x -8/3)^2 -64/3 - y² +20=0.Convert 20 to thirds: 20 = 60/3.So:3(x -8/3)^2 - y² -64/3 +60/3 =0Which is:3(x -8/3)^2 - y² -4/3=0Then move the constant:3(x -8/3)^2 - y² =4/3Divide both sides by 4/3 to get 1 on the right:[3(x -8/3)^2]/(4/3) - y²/(4/3) =1Which simplifies to:[ (x -8/3)^2 ]/(4/9) - [ y² ]/(4/3) =1Which is:(x -8/3)^2 / (4/9) - y²/(4/3) =1Expressed as:[(x - 8/3)^2]/( (2/3)^2 ) - [y^2]/( (2/√3)^2 ) =1So this is a hyperbola centered at (8/3, 0), with a horizontal transverse axis. But none of the answer choices A or B are in this form. A and B are centered at the origin. Therefore, the answer must be option C, which is the equation we derived before completing the square. The original equation is C. So the answer is C. Let me check if C is equivalent to the standard form I found.Wait, let me check the answer choices again. Option C is 3x² - y² -16x +20=0, which is exactly what I derived. So perhaps the correct answer is C. However, the answer options A and B are standard hyperbola equations, but they are not shifted, which would be necessary if the center is at (8/3,0). Therefore, since the problem is from a competition, they might have given the equation in general form rather than standard form, so C is the correct answer. But let me check if maybe there's a miscalculation.Alternatively, maybe I can cross-validate by plugging in a point. Let's see. Suppose x=4, which is the focus. Let's see what y would be. Plug x=4 into the equation C: 3*(16) - y² -16*4 +20= 48 - y² -64 +20= (48 -64 +20) - y²=4 - y²=0, so y²=4, so y=±2. Let's check if this point (4,2) satisfies the original ratio condition.Distance to F(4,0): sqrt((4-4)^2 + (2-0)^2)=sqrt(0 +4)=2.Distance to line x=3 is |4 -3|=1.Ratio is 2/1=2, which matches. So (4,2) is on the locus and satisfies the condition, and also lies on equation C. Similarly, another point. Let's take x=8/3, the center. Plug x=8/3 into equation C:3*(64/9) - y² -16*(8/3) +20= 64/3 - y² -128/3 +20= (-64/3 - y²) +20= -y² -64/3 +60/3= -y² -4/3=0 ⇒ -y²=4/3 ⇒ y²= -4/3. Which is impossible, so at x=8/3, there are no real points, which makes sense for a hyperbola. The vertices would be at (8/3 ± a, 0). From the standard form I found earlier, a²=4/9, so a=2/3. So vertices at (8/3 +2/3, 0)=(10/3,0) and (8/3 -2/3, 0)=(6/3,0)=(2,0). Let's check (2,0). Plug into equation C: 3*(4) -0 -16*2 +20=12 -32 +20=0. So (2,0) is on the locus. Check the ratio: distance to F(4,0) is sqrt((2-4)^2 +0)=2. Distance to line x=3 is |2-3|=1. Ratio 2/1=2. Correct. Similarly, (10/3,0): Plug into equation C: 3*(100/9) -0 -16*(10/3) +20= 100/3 -160/3 +20= (-60/3) +20= -20 +20=0. So (10/3,0) is also on the locus. Check the ratio: distance to F(4,0): sqrt((10/3 -4)^2 +0)=sqrt((-2/3)^2)=2/3. Distance to line x=3: |10/3 -3|=|1/3|=1/3. Ratio (2/3)/(1/3)=2. Correct. So all checks out. Therefore, the answer must be C. But the options A and B are standard hyperbola forms. Let me see why they are different.If we consider standard hyperbola forms centered at the origin, but our hyperbola is centered at (8/3,0). Therefore, the answer is not A or B. Therefore, the correct answer is C.Final Answerboxed{C}</think>Given a moving point ( M(x, y) ) such that the ratio of its distance to the point ( F(4, 0) ) to its distance to the line ( x = 3 ) is 2, we need to find the equation of the locus of ( M(x, y) ).1. Distance to Point ( F(4, 0) ): [ sqrt{(x - 4)^2 + y^2} ]2. Distance to Line ( x = 3 ): [ |x - 3| ]3. Given Ratio: [ frac{sqrt{(x - 4)^2 + y^2}}{|x - 3|} = 2 ]4. Square Both Sides: [ (x - 4)^2 + y^2 = 4(x - 3)^2 ]5. Expand Both Sides: - Left side: ( (x - 4)^2 + y^2 = x^2 - 8x + 16 + y^2 ) - Right side: ( 4(x - 3)^2 = 4(x^2 - 6x + 9) = 4x^2 - 24x + 36 )6. Set Up Equation: [ x^2 - 8x + 16 + y^2 = 4x^2 - 24x + 36 ]7. Combine Like Terms: [ x^2 - 8x + 16 + y^2 - 4x^2 + 24x - 36 = 0 ] [ -3x^2 + 16x - 20 + y^2 = 0 ]8. Rearrange Terms: [ 3x^2 - y^2 - 16x + 20 = 0 ]Thus, the equation of the locus of ( M(x, y) ) is ( 3x^2 - y^2 - 16x + 20 = 0 ), which corresponds to option C.[boxed{C}]]

🔑:[Solution] According to the problem, we have frac{sqrt{(dot{x}-4)^{2}+y^{2}}}{|x-3|}=2, which means3 x^{2}-y^{2}-16 x+20=0 text {. }Therefore, the correct choice is (C).

🧠:[<think>Alright, let's tackle this problem step by step. It's about a triangle ABC with sides a, b, c opposite angles A, B, C respectively. The first part asks to find angle C given the equation (b - 2a)·cos C + c·cos B = 0. The second part gives c = 2 and area sqrt(3), and wants us to find a and b.Starting with part (1). The equation given is (b - 2a)cos C + c cos B = 0. Hmm, this seems to involve trigonometric identities and perhaps the Law of Cosines or Sines. Let me recall those. The Law of Cosines states that c² = a² + b² - 2ab cos C, and similarly for the other sides and angles. The Law of Sines says a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.But here we have cosines of angles multiplied by sides. Maybe we can use the projection formula? Because in a triangle, the projection of one side onto another can be expressed in terms of the other sides and the cosine of included angles. For example, the projection of side c onto side a would be b cos A + c cos B = a. Wait, that seems familiar. Let me check.Yes, in any triangle, the length of a side is equal to the sum of the projections of the other two sides onto it. So, for example, a = b cos C + c cos B. Similarly, b = a cos C + c cos A, and c = a cos B + b cos A. These are known as the projection formulas.So, given that, let's see. The given equation is (b - 2a) cos C + c cos B = 0. Let's expand that: b cos C - 2a cos C + c cos B = 0. Let's rearrange the terms: (b cos C + c cos B) - 2a cos C = 0. But from the projection formula, we know that a = b cos C + c cos B. Therefore, substituting into the equation: a - 2a cos C = 0. So that simplifies to a(1 - 2 cos C) = 0.Since a is a side length of a triangle, it cannot be zero. Therefore, we must have 1 - 2 cos C = 0. Solving for cos C gives cos C = 1/2. Therefore, angle C is arccos(1/2), which is 60 degrees or π/3 radians.Wait, that seems straightforward. Let me verify. The key step was recognizing the projection formula. If we have (b cos C + c cos B) which equals a, then substituting that into the equation gives a - 2a cos C = 0, leading to cos C = 1/2. Yes, that seems correct. Therefore, angle C is 60 degrees. So part (1) answer is π/3 or 60°.Moving on to part (2). Given c = 2 and area S = sqrt(3). Need to find a and b. Since we already know angle C is 60°, which we found in part (1). So, in triangle ABC, angle C is 60°, side c = 2, area sqrt(3). Let's recall that the area of a triangle can be calculated using the formula (1/2)ab sin C. Wait, but here sides a and b are adjacent to angle C? Wait, in triangle ABC, sides a, b, c are opposite angles A, B, C respectively. So angle C is opposite side c. So sides a and b are adjacent to angle C? Wait, angle C is between sides a and b. Wait, no. Let me clarify.In triangle ABC, side a is opposite angle A, side b opposite angle B, and side c opposite angle C. Therefore, angle C is between sides a and b. So the two sides forming angle C are a and b, and the side opposite angle C is c. So, the area of the triangle can be expressed as (1/2)*a*b*sin C. That's correct.Given that, and since we know angle C is 60°, sin 60° = sqrt(3)/2. So the area S = (1/2)*a*b*(sqrt(3)/2) = (sqrt(3)/4)*a*b. And we are given that this area is sqrt(3). Therefore:(sqrt(3)/4)*a*b = sqrt(3)Divide both sides by sqrt(3):(1/4)*a*b = 1Thus, a*b = 4.So we have equation (1): a*b = 4.Additionally, since we know side c = 2, and angle C = 60°, we can use the Law of Cosines to relate a, b, and c. The Law of Cosines for angle C is:c² = a² + b² - 2ab cos CSubstituting c = 2, cos C = 1/2:2² = a² + b² - 2ab*(1/2)Simplify:4 = a² + b² - abTherefore, equation (2): a² + b² - ab = 4So now, we have two equations:1. a*b = 42. a² + b² - a*b = 4Let me substitute equation 1 into equation 2. Since a*b = 4, then equation 2 becomes:a² + b² - 4 = 4 => a² + b² = 8So now, we have:a² + b² = 8anda*b = 4We need to solve for a and b. Let's consider these two equations. Let me recall that (a + b)² = a² + 2ab + b², and (a - b)² = a² - 2ab + b².Given that a² + b² = 8 and ab = 4, then:(a + b)² = 8 + 2*4 = 16 => a + b = 4 (since lengths are positive)Similarly, (a - b)² = 8 - 2*4 = 0 => a - b = 0 => a = bWait, that's interesting. If (a - b)² = 0, then a = b. So the triangle is isoceles with a = b. Then, since a = b and angle C = 60°, the triangle would be equilateral? Wait, because if two sides are equal and the angle between them is 60°, then all sides would be equal.Wait, but c = 2. If a = b, and angle C = 60°, then by the Law of Cosines:c² = a² + a² - 2a²*(1/2) => 4 = 2a² - a² => 4 = a² => a = 2. Therefore, a = b = 2. Then c = 2. So the triangle is equilateral with all sides equal to 2. But then, the area would be (sqrt(3)/4)*(2)^2 = sqrt(3), which matches the given area. So that works.Therefore, the solution is a = 2 and b = 2.Wait, but let's confirm step by step. From the equations:a + b = 4 and a - b = 0. So solving these two equations:From a - b = 0 => a = b.Substitute into a + b = 4: a + a = 4 => 2a = 4 => a = 2, so b = 2.Yes, so a = b = 2. Therefore, sides a and b are both 2. Since angle C is 60°, sides a and b are both 2, and side c is 2, so it's an equilateral triangle. The area is sqrt(3), which matches an equilateral triangle with side 2: (sqrt(3)/4)*2² = sqrt(3). Correct.Therefore, the values of a and b are both 2.So summarizing:(1) Angle C is 60°, so π/3 radians.(2) Sides a and b are both 2.I think that's the solution. Let me check if there could be another possibility. For part (2), we had two equations leading to a = b = 2. Since the quadratic equations lead to only this solution, I don't think there's another set of a and b that satisfies both the area and the Law of Cosines. Therefore, the answer is a = 2, b = 2.Final Answer(1) The measure of angle ( C ) is (boxed{dfrac{pi}{3}}).(2) The lengths of sides ( a ) and ( b ) are both (boxed{2}).</think>Given triangle (ABC) with sides (a), (b), and (c) opposite angles (A), (B), and (C) respectively, we need to solve the following:1. Find the angle (C) given the equation ((b - 2a) cos C + c cos B = 0).2. Find the values of (a) and (b) given (c = 2) and the area (S_{triangle ABC} = sqrt{3}). Part (1)We start with the given equation:[(b - 2a) cos C + c cos B = 0]Expanding and rearranging terms:[b cos C - 2a cos C + c cos B = 0]Using the projection formula (a = b cos C + c cos B), we substitute (a) into the equation:[a - 2a cos C = 0]This simplifies to:[a(1 - 2 cos C) = 0]Since (a neq 0), we have:[1 - 2 cos C = 0 implies cos C = frac{1}{2}]Thus, angle (C) is:[C = arccosleft(frac{1}{2}right) = frac{pi}{3}] Part (2)Given (c = 2) and the area (S_{triangle ABC} = sqrt{3}), we use the area formula for a triangle:[S_{triangle ABC} = frac{1}{2}ab sin C]Substituting (sin frac{pi}{3} = frac{sqrt{3}}{2}) and the given area:[sqrt{3} = frac{1}{2}ab cdot frac{sqrt{3}}{2} implies sqrt{3} = frac{sqrt{3}}{4}ab]Solving for (ab):[ab = 4]Using the Law of Cosines for angle (C):[c^2 = a^2 + b^2 - 2ab cos C]Substituting (c = 2), (cos frac{pi}{3} = frac{1}{2}):[4 = a^2 + b^2 - 2ab cdot frac{1}{2} implies 4 = a^2 + b^2 - ab]We already know (ab = 4), so:[4 = a^2 + b^2 - 4 implies a^2 + b^2 = 8]We solve the system of equations:[a^2 + b^2 = 8 quad text{and} quad ab = 4]Since ((a + b)^2 = a^2 + 2ab + b^2):[(a + b)^2 = 8 + 8 = 16 implies a + b = 4]And ((a - b)^2 = a^2 - 2ab + b^2):[(a - b)^2 = 8 - 8 = 0 implies a = b]Thus, (a = b = 2). Final Answers1. The measure of angle (C) is (boxed{dfrac{pi}{3}}).2. The lengths of sides (a) and (b) are both (boxed{2}).,<think>Okay, let me try to work through this problem step by step. It's about triangle ABC with sides a, b, c opposite angles A, B, C respectively. The given equation is (b - 2a) * cos C + c * cos B = 0. There are two parts: first, find angle C; second, given c=2 and area sqrt(3), find sides a and b.Starting with part (1). The equation given is (b - 2a) cos C + c cos B = 0. Hmm. I need to find angle C. I remember that in triangles, the Law of Cosines relates sides and angles. Also, maybe the Law of Sines. Let me recall those.Law of Cosines: c² = a² + b² - 2ab cos C. Similarly for other angles. Law of Sines: a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.Alternatively, there's the projection formula: in any triangle, the length of a side can be expressed as the sum of the projections of the other two sides onto it. For example, c = a cos B + b cos A. Maybe that's useful here since the given equation involves terms like cos C and cos B multiplied by sides.Wait, the given equation is (b - 2a) cos C + c cos B = 0. Let's expand that: b cos C - 2a cos C + c cos B = 0. If I rearrange terms: b cos C + c cos B = 2a cos C.But from the projection formula, I know that in triangle ABC, the side a is equal to b cos C + c cos B. Let me verify that. Yes, because if you project sides b and c onto side a, their projections should add up to a. So, a = b cos C + c cos B. So the left side of the equation (b cos C + c cos B) is equal to a. Therefore, substituting that into the equation gives a = 2a cos C. So a = 2a cos C.Subtracting 2a cos C from both sides gives a - 2a cos C = 0. Factor out a: a(1 - 2 cos C) = 0. Since a is a side length of a triangle, it can't be zero. Therefore, 1 - 2 cos C = 0. Solving for cos C gives cos C = 1/2. Therefore, angle C is 60 degrees or π/3 radians. So answer to part (1) is C = 60°, or π/3.Let me check if this makes sense. If cos C = 1/2, then C = 60°, which is a common angle in triangles. The steps seem logical. Using the projection formula to substitute into the given equation simplifies things directly. So part (1) seems solved.Moving on to part (2). Given c = 2, and area S = sqrt(3). Need to find a and b. Since we already know angle C is 60°, we can use the area formula involving sides a, b, and angle C. The area S = (1/2)ab sin C. Since C is 60°, sin C = sqrt(3)/2. Therefore, S = (1/2)ab*(sqrt(3)/2) = (ab sqrt(3))/4. Given that S = sqrt(3), so:(ab sqrt(3))/4 = sqrt(3)Multiply both sides by 4:ab sqrt(3) = 4 sqrt(3)Divide both sides by sqrt(3):ab = 4So, ab = 4. That's one equation. Now, we need another equation to relate a and b. Since we know angle C = 60°, we can use the Law of Cosines for side c, which is opposite angle C. Law of Cosines says:c² = a² + b² - 2ab cos CWe know c = 2, angle C = 60°, so cos C = 1/2. Substituting:2² = a² + b² - 2ab*(1/2)Simplify:4 = a² + b² - abSo we have:a² + b² - ab = 4And from the area, we have ab = 4.So now, we have two equations:1) ab = 42) a² + b² - ab = 4Let me substitute ab = 4 into the second equation:a² + b² - 4 = 4Therefore,a² + b² = 8So now, we have a² + b² = 8 and ab = 4.Let me think how to solve for a and b. Let me denote x = a + b and y = a - b. Alternatively, since we have a² + b² and ab, we can use the identity (a + b)² = a² + 2ab + b². We know a² + b² = 8 and ab = 4. Therefore,(a + b)² = 8 + 2*4 = 8 + 8 = 16Therefore, a + b = sqrt(16) = 4. Since sides are positive, a + b = 4.Similarly, (a - b)² = a² - 2ab + b² = (a² + b²) - 2ab = 8 - 8 = 0Therefore, (a - b)^2 = 0 => a - b = 0 => a = bSo if a + b = 4 and a = b, then 2a = 4 => a = 2. Then b = 2.So both a and b are 2. Let me check if this satisfies the equations.ab = 2*2 = 4. Correct.a² + b² = 4 + 4 = 8. Correct.Law of Cosines: c² = 4 = 4 + 4 - 2*2*2*(1/2) = 8 - 4 = 4. Correct.Area: (1/2)*2*2*(sqrt(3)/2) = (1/2)*4*(sqrt(3)/2) = (2)*(sqrt(3)/2) = sqrt(3). Correct.Therefore, the sides a and b are both 2. So answer to part (2) is a = 2, b = 2.Wait, but is this possible? If all sides are 2, then it's an equilateral triangle. Since angle C is 60°, if all sides are equal, all angles are 60°, so that makes sense. So the triangle is equilateral. But in the problem, c=2, so all sides are 2. Therefore, a = b = 2.Alternatively, could there be another solution where a ≠ b?But according to the equations, (a - b)^2 = 0, so a = b is the only solution. Therefore, the triangle is equilateral. Hence, sides a and b are both 2. So that seems correct.Therefore, final answers: (1) angle C is 60°, and (2) a = 2, b = 2.Final Answer(1) The angle ( C ) is (boxed{dfrac{pi}{3}}).(2) The sides are ( a = boxed{2} ) and ( b = boxed{2} ).</think>Given triangle (ABC) with sides (a), (b), and (c) opposite angles (A), (B), and (C) respectively, we know the equation ((b - 2a) cos C + c cos B = 0).(1) To find the angle (C):1. Start with the given equation: ((b - 2a) cos C + c cos B = 0).2. Expand and rearrange terms: (b cos C + c cos B = 2a cos C).3. Using the projection formula, we know (a = b cos C + c cos B). Substituting this into the equation gives (a = 2a cos C).4. Solving for (cos C): (1 - 2 cos C = 0 implies cos C = frac{1}{2}).5. Therefore, (C = frac{pi}{3}) (60 degrees).(2) Given (c = 2) and the area (S_{triangle ABC} = sqrt{3}):1. The area formula using sides (a), (b), and angle (C): (S = frac{1}{2}ab sin C).2. Substituting the known values: (sqrt{3} = frac{1}{2}ab cdot frac{sqrt{3}}{2} implies ab = 4).3. Using the Law of Cosines for (c): (c^2 = a^2 + b^2 - 2ab cos C).4. Substituting (c = 2) and (cos C = frac{1}{2}): (4 = a^2 + b^2 - ab).5. We have two equations: (ab = 4) and (a^2 + b^2 - ab = 4).6. Substituting (ab = 4) into the second equation: (a^2 + b^2 = 8).7. Solving these equations, we find (a = 2) and (b = 2) (since (a = b) and (a + b = 4)).Final answers:(1) The angle (C) is (boxed{dfrac{pi}{3}}).(2) The sides are (a = boxed{2}) and (b = boxed{2}).]

🔑:(1) Given that (b - 2a) cdot cos C + c cdot cos B = 0, by the Law of Sines, we can rewrite the equation as:(b - 2a) cdot frac{sin C}{sin A} + frac{sin B cdot c}{sin A} = 0.This simplifies to (sin B - 2 sin A) cos C + sin C cos B = 0.Hence, we have sin B cos C + cos B sin C = 2 sin A cos C, which impliessin (B + C) = sin A = 2 sin A cos C.Since sin A neq 0, we have cos C = frac{1}{2}.Given that angle C is within the range (0, pi), angle C must be boxed{frac{pi}{3}}.(2) The area S_{triangle ABC} is given by frac{1}{2}absin C. Plugging in the given area sqrt{3} and sin C = sin frac{pi}{3} = frac{sqrt{3}}{2}, we get:frac{1}{2}absin C = frac{sqrt{3}}{4}ab = sqrt{3}.Therefore, ab = 4. This is our equation (1).Using the Law of Cosines, we have a^2 + b^2 - c^2 = 2abcos C. With c = 2, C = frac{pi}{3}, and ab = 4, this yields:a^2 + b^2 = 8. This is our equation (2).Solving equations (1) ab = 4 and (2) a^2 + b^2 = 8 simultaneously, we find a = 2 and b = 2. Thus, the lengths of the sides a and b are boxed{a = 2} and boxed{b = 2} respectively.

🧠:[<think>Okay, so I have this geometry problem here. Let me read it again: "BL is the bisector of triangle ABC. Find its area, given that |AL|=3, |BL|=6√5, |CL|=4." Hmm, okay. So BL is the bisector. Wait, angle bisector or side bisector? The problem says "bisector of triangle ABC." Hmm, usually when they mention a bisector in a triangle without specifying, it's often an angle bisector. Also, the segments given are AL, BL, and CL. Let me visualize this.So triangle ABC, with point L somewhere on side AC, since BL is the bisector. Wait, if BL is the angle bisector, then it should split the opposite side into segments proportional to the adjacent sides. So angle bisector theorem? Yeah, that's right. The angle bisector theorem states that if a bisector of an angle of a triangle divides the opposite side into segments proportional to the adjacent sides. So in this case, if BL is the angle bisector of angle B, then AL / LC = AB / BC.But here they give the lengths of AL, BL, and CL. Let's note that:- AL = 3- CL = 4- BL = 6√5So since AL and CL are parts of side AC, then AC = AL + CL = 3 + 4 = 7.And by the angle bisector theorem, AL / CL = AB / BC. So 3/4 = AB / BC. So AB = (3/4) BC. Let me note that AB = (3/4) BC. So if I let BC = x, then AB = (3/4)x.But how does this help me find the area? Hmm. Maybe I need to use some formula that relates the lengths of the angle bisector to the sides of the triangle and then use that to find the area.I recall that there is a formula for the length of an angle bisector in a triangle. Let me recall it. The formula is:The length of the angle bisector from angle B to side AC is given by:BL = (2 * AB * BC * cos(theta/2)) / (AB + BC)Wait, or another formula. Alternatively, the formula is:BL^2 = AB * BC - AL * LCWait, is that right? Let me check. If BL is the angle bisector, then there is a relation called the angle bisector formula which relates the length of the bisector to the sides.Yes, Stewart's theorem! Stewart's theorem can be used here. Stewart's theorem relates the lengths of the sides of a triangle to the length of a cevian. In this case, BL is the cevian.Stewart's theorem states that for a triangle with sides a, b, c, and a cevian of length d to the side a, which divides it into segments m and n, the theorem is:b^2 m + c^2 n = a(d^2 + mn)So applying that here. Let's assign the sides. Let me label the triangle properly.In triangle ABC, BL is the angle bisector from B to side AC. So:- Side AC is divided by L into AL = 3 and CL = 4.Therefore, AC = 7.Let me denote:- AB = c- BC = a- AC = b = 7- The angle bisector BL has length d = 6√5Then according to the angle bisector theorem, AL / CL = AB / BC => 3/4 = c / a => c = (3/4)aNow, applying Stewart's theorem:AB² * CL + BC² * AL = AC * (BL² + AL * CL)Plugging in the known values:c² * 4 + a² * 3 = 7 * [(6√5)^2 + 3*4]First, let's compute (6√5)^2: that's 36 * 5 = 180And 3*4 = 12So the right side becomes 7*(180 + 12) = 7*192 = 1344Left side: c² *4 + a² *3. But we know that c = (3/4)a, so c² = (9/16)a². Therefore:(9/16)a² *4 + a² *3 = (9/4)a² + 3a² = (9/4 + 12/4)a² = (21/4)a²So (21/4)a² = 1344Multiply both sides by 4: 21a² = 5376Divide by 21: a² = 5376 / 21Compute 5376 ÷ 21. Let's see:21*256 = 5376 (since 21*200=4200, 21*50=1050, 21*6=126; 4200+1050=5250, 5250+126=5376)So a² = 256 => a = 16Therefore, BC = a = 16Then AB = (3/4)a = (3/4)*16 = 12So sides AB = 12, BC = 16, AC = 7.Wait, but hold on. If AB = 12, BC = 16, and AC = 7, does that satisfy the triangle inequality?Check AB + BC > AC: 12 + 16 = 28 > 7, okay.AB + AC > BC: 12 + 7 = 19 > 16, okay.BC + AC > AB: 16 + 7 = 23 > 12, okay. So that's fine.Now, we need to find the area of triangle ABC. The sides are 12, 16, 7. Hmm. Let's see. Since we have all three sides, we can use Heron's formula.Heron's formula: area = √[s(s - a)(s - b)(s - c)], where s is the semi-perimeter.First, compute the semi-perimeter:s = (12 + 16 + 7)/2 = (35)/2 = 17.5Then area = √[17.5*(17.5 - 12)*(17.5 - 16)*(17.5 - 7)]Compute each term:17.5 - 12 = 5.517.5 - 16 = 1.517.5 - 7 = 10.5So area = √[17.5 * 5.5 * 1.5 * 10.5]Hmm, this might be a bit tedious, but let's compute it step by step.First, note that 17.5 = 35/2, 5.5 = 11/2, 1.5 = 3/2, 10.5 = 21/2So multiplying them together:(35/2) * (11/2) * (3/2) * (21/2) = (35 * 11 * 3 * 21) / (2^4) = (35 * 11 * 63) / 16Compute numerator:35 * 11 = 385385 * 63: Let's compute that.385 * 60 = 23100385 * 3 = 115523100 + 1155 = 24255So numerator is 24255Denominator is 16Therefore, the product is 24255 / 16So area = √(24255 / 16) = √(24255)/4But wait, let's see if 24255 can be simplified.Factor 24255:Divide by 5: 24255 ÷5=48514851 ÷ 3=16171617 ÷ 3=539539 ÷7=7777 ÷7=11So prime factors: 5 * 3^2 * 7^2 * 11Therefore, √(24255) = √(5 * 9 * 49 * 11) = √(9 * 49 * 55) = 3 *7 * √55 = 21√55Therefore, area = 21√55 /4Wait, but let me check that again:Wait, 24255 = 5 * 3^2 *7^2 *11, so sqrt(24255) = 3*7*sqrt(5*11) = 21√55Yes. So area is 21√55 /4. Hmm. That seems like a possible answer, but let me see if there's another way.Alternatively, maybe using the formula involving the angle bisector and the area.Alternatively, maybe using coordinates. Let me try that to verify.Let me place point A at the origin (0,0), point C at (7,0) since AC is 7 units long. Then point L is located 3 units from A and 4 units from C, so at (3,0). Wait, but BL is the angle bisector. Wait, but in that case, point B is somewhere in the plane, and BL connects B to L(3,0), with BL = 6√5. Hmm.So coordinates: Let me set coordinates with A(0,0), C(7,0), L(3,0). Then point B is somewhere above the x-axis. Let me denote point B as (x,y). Then, we have distances:From B to L(3,0) is 6√5, so sqrt[(x -3)^2 + (y -0)^2] = 6√5, so (x -3)^2 + y^2 = 36*5 = 180.Also, AB = 12, so distance from A(0,0) to B(x,y) is sqrt(x^2 + y^2) = 12, so x^2 + y^2 = 144.Similarly, BC = 16, so distance from B(x,y) to C(7,0) is sqrt[(x -7)^2 + y^2] = 16, so (x -7)^2 + y^2 = 256.So now we have three equations:1. (x -3)^2 + y^2 = 1802. x^2 + y^2 = 1443. (x -7)^2 + y^2 = 256Let's subtract equation 2 from equation 1:[(x -3)^2 + y^2] - [x^2 + y^2] = 180 - 144 => (x^2 -6x +9) -x^2 = 36 => -6x +9 = 36 => -6x = 27 => x = -27/6 = -4.5 = -9/2Wait, x = -9/2? So that's x = -4.5. Hmm. Let's check if this is consistent with the other equations.So if x = -9/2, then from equation 2: x^2 + y^2 = 144 => (81/4) + y^2 = 144 => y^2 = 144 - 81/4 = (576/4 -81/4) = 495/4 => y = ±√(495/4) = ±(√495)/2. Let's keep it as y = √(495)/2 for simplicity, since the triangle is above the x-axis.Now check equation 3: (x -7)^2 + y^2 should equal 256.Compute (x -7)^2 + y^2:x = -9/2, so x -7 = (-9/2 -14/2) = (-23/2). So squared is (529/4)y^2 = 495/4So total: 529/4 + 495/4 = (529 + 495)/4 = 1024/4 = 256. Perfect, that checks out.So coordinates of B are (-9/2, √495/2). Then, the area of triangle ABC can be computed using the coordinates.Since points A(0,0), C(7,0), B(-9/2, √495/2). The area can be calculated via the shoelace formula.Formula:Area = 1/2 | (x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)) |Plugging in:x_A = 0, y_A = 0x_B = -9/2, y_B = √495/2x_C = 7, y_C = 0So:Area = 1/2 | 0*(√495/2 - 0) + (-9/2)*(0 - 0) +7*(0 - √495/2) |Simplify:= 1/2 | 0 + 0 +7*(-√495/2) | = 1/2 | -7√495/2 | = 1/2 * (7√495/2) = 7√495 /4But √495 can be simplified. 495 = 9*55 = 9*5*11. So √495 = 3√55Therefore, Area = 7*(3√55)/4 = 21√55 /4Same result as before. So that's good, confirms the answer.But let me just think again: Heron's formula gave 21√55 /4, and coordinate method also gave the same. So seems correct.Alternatively, another method using the formula for area in terms of the angle bisector.Wait, but maybe another approach using trigonometry. Let's see.We know sides AB=12, BC=16, AC=7. Let me compute the angles using the Law of Cosines, and then compute the area as (1/2)*ab*sin(theta).But maybe that's more complicated. Let's check.Compute angle at B, using Law of Cosines:cos B = (AB² + BC² - AC²)/(2*AB*BC) = (12² +16² -7²)/(2*12*16) = (144 +256 -49)/384 = (351)/384 = 117/128Therefore, sin B = sqrt(1 - (117/128)^2). Let's compute that.(117/128)^2 = (13689)/16384So 1 - 13689/16384 = (16384 -13689)/16384 = 2695/16384Therefore, sin B = sqrt(2695/16384) = sqrt(2695)/128But 2695 factors: 2695 ÷5=539. 539 ÷7=77. 77 ÷7=11. So 2695=5*7*7*11=5*7²*11. Therefore sqrt(2695)=7*sqrt(55). Therefore, sin B=7*sqrt(55)/128Then area = (1/2)*AB*BC*sin B = (1/2)*12*16*(7√55)/128Compute that:12*16=192192*(7√55)/ (2*128) = (192*7√55)/256Simplify 192/256 = 3/4So 3/4 *7√55 =21√55 /4Same result. So all methods lead to 21√55 /4. So that must be the correct answer.Therefore, the area of triangle ABC is 21√55 divided by 4, so boxed answer is boxed{dfrac{21sqrt{55}}{4}}.Final Answerboxed{dfrac{21sqrt{55}}{4}}</think>Given triangle (ABC) with (BL) as the angle bisector, we need to find its area given (|AL| = 3), (|BL| = 6sqrt{5}), and (|CL| = 4).1. Applying the Angle Bisector Theorem: - By the angle bisector theorem, (frac{AL}{LC} = frac{AB}{BC}). - Given (AL = 3) and (CL = 4), we have (frac{3}{4} = frac{AB}{BC}), leading to (AB = frac{3}{4}BC).2. Using Stewart's Theorem: - Stewart's theorem states (AB^2 cdot CL + BC^2 cdot AL = AC cdot (BL^2 + AL cdot CL)). - Substituting the given values: [ left(frac{3}{4}BCright)^2 cdot 4 + BC^2 cdot 3 = 7 cdot left((6sqrt{5})^2 + 3 cdot 4right) ] - Simplifying the right-hand side: [ 7 cdot (180 + 12) = 7 cdot 192 = 1344 ] - Simplifying the left-hand side: [ left(frac{9}{16}BC^2right) cdot 4 + 3BC^2 = frac{9}{4}BC^2 + 3BC^2 = frac{21}{4}BC^2 ] - Solving for (BC): [ frac{21}{4}BC^2 = 1344 implies BC^2 = 256 implies BC = 16 ] - Therefore, (AB = frac{3}{4} cdot 16 = 12).3. Using Heron's Formula: - The sides of the triangle are (AB = 12), (BC = 16), and (AC = 7). - The semi-perimeter (s) is: [ s = frac{12 + 16 + 7}{2} = 17.5 ] - The area is calculated as: [ text{Area} = sqrt{s(s - a)(s - b)(s - c)} = sqrt{17.5 times 5.5 times 1.5 times 10.5} ] - Converting to fractions and simplifying: [ sqrt{frac{35}{2} times frac{11}{2} times frac{3}{2} times frac{21}{2}} = sqrt{frac{24255}{16}} = frac{21sqrt{55}}{4} ]Thus, the area of triangle (ABC) is (boxed{dfrac{21sqrt{55}}{4}}).,<think>Okay, so I have this geometry problem here. Triangle ABC, and BL is the bisector. They want me to find the area of the triangle. The given lengths are AL = 3, BL = 6√5, and CL = 4. Hmm, let me visualize this first. BL is the bisector, so point L must be on side AC, right? Because an angle bisector in a triangle goes from a vertex to the opposite side, dividing the angle into two equal parts. So, in this case, BL is the bisector of angle B, splitting side AC into segments AL and LC with lengths 3 and 4 respectively. So AL is 3, LC is 4, meaning AC is 3 + 4 = 7 units long.Now, since BL is the angle bisector, there's a theorem about angle bisectors that relates the lengths of the sides. The Angle Bisector Theorem states that the ratio of the two segments created by the bisector on the opposite side is equal to the ratio of the other two sides of the triangle. So, in this case, the ratio of AL to LC should be equal to the ratio of AB to BC. Let me write that down:AL / LC = AB / BCGiven AL = 3 and LC = 4, so 3/4 = AB / BC. That gives me AB = (3/4) * BC. Let me note that AB = (3/4)BC. Maybe I can use this later.But how does this help me find the area? I need more information. I also know the length of BL, which is 6√5. Maybe there's a formula related to the length of an angle bisector in a triangle? Let me recall that.Yes, there is a formula for the length of an angle bisector. The length of the angle bisector from angle B to side AC is given by:BL = (2AB * BC * cos(theta/2)) / (AB + BC)Where theta is the measure of angle B. Hmm, but I don't know theta. Alternatively, another formula for the length of the angle bisector is:BL^2 = AB * BC - AL * LCWait, let me check that. Let me recall Stewart's Theorem. Stewart's Theorem relates the lengths of the sides of a triangle to the length of a cevian. In this case, BL is a cevian. Stewart's Theorem states:AB^2 * LC + BC^2 * AL = AC * (BL^2 + AL * LC)Yes, that seems right. Let me plug in the known values here.Given AL = 3, LC = 4, AC = 7, BL = 6√5. So substituting into Stewart's Theorem:AB^2 * 4 + BC^2 * 3 = 7 * ((6√5)^2 + 3*4)First, compute (6√5)^2. That's 36 * 5 = 180. Then 3*4 = 12. So inside the parentheses, 180 + 12 = 192. Therefore, the right-hand side is 7 * 192 = 1344.So the equation becomes:4AB^2 + 3BC^2 = 1344But from the Angle Bisector Theorem, earlier, we have AB = (3/4)BC. Let me substitute AB in terms of BC into the equation.Let AB = (3/4)BC. Then AB^2 = (9/16)BC^2. Substituting into the equation:4*(9/16)BC^2 + 3BC^2 = 1344Simplify the terms:First term: 4*(9/16) = (36/16) = (9/4)So (9/4)BC^2 + 3BC^2 = 1344Convert 3BC^2 to fourths: 3BC^2 = (12/4)BC^2Adding together: (9/4 + 12/4)BC^2 = (21/4)BC^2 = 1344Therefore, BC^2 = 1344 * (4/21) = (1344/21)*4Calculate 1344 divided by 21: 21*64 = 1344, so 1344/21 = 64Therefore, BC^2 = 64 * 4 = 256Thus, BC = √256 = 16Then AB = (3/4) * BC = (3/4)*16 = 12So now we know sides AB = 12, BC = 16, and AC = 7. Wait, hold on, AC was given as AL + LC = 3 + 4 = 7. So the sides of the triangle are AB = 12, BC = 16, AC = 7.But wait, if the sides are 12, 16, and 7, does that make sense? Let me check the triangle inequality. The sum of AB and AC should be greater than BC: 12 + 7 = 19 > 16. The sum of BC and AC: 16 + 7 = 23 > 12. The sum of AB and BC: 12 + 16 = 28 > 7. So yes, the triangle inequalities are satisfied. So the sides are 12, 16, 7.Now, the problem is to find the area of triangle ABC. With all three sides known, we can use Heron's formula. Heron's formula states that the area is sqrt[s(s - a)(s - b)(s - c)], where s is the semi-perimeter.First, compute the semi-perimeter, s:s = (AB + BC + AC)/2 = (12 + 16 + 7)/2 = (35)/2 = 17.5Then the area is sqrt[17.5*(17.5 - 12)*(17.5 - 16)*(17.5 - 7)]Compute each term:17.5 - 12 = 5.517.5 - 16 = 1.517.5 - 7 = 10.5So the area is sqrt[17.5 * 5.5 * 1.5 * 10.5]Hmm, that looks a bit messy. Let me compute this step by step.First, multiply 17.5 and 10.5:17.5 * 10.5 = (17.5)*(10 + 0.5) = 17.5*10 + 17.5*0.5 = 175 + 8.75 = 183.75Next, multiply 5.5 and 1.5:5.5 * 1.5 = (5 + 0.5)*(1 + 0.5) = 5*1 + 5*0.5 + 0.5*1 + 0.5*0.5 = 5 + 2.5 + 0.5 + 0.25 = 8.25Now multiply the two products: 183.75 * 8.25Hmm, this is going to be a bit tedious. Let me compute 183.75 * 8.25.First, note that 8.25 is 8 + 0.25. So:183.75 * 8 = 1470183.75 * 0.25 = 45.9375Adding these together: 1470 + 45.9375 = 1515.9375Therefore, the area is sqrt(1515.9375). Let me compute that.First, note that 1515.9375 is equal to 1515.9375 = 15159375/10000. But maybe there's a better way. Alternatively, let's see if this number is a perfect square or can be simplified.Alternatively, perhaps there's a mistake here. Because getting such a complicated number seems odd. Let me check my calculations again.Wait, maybe instead of using Heron's formula, there's another way using the angle bisector and the lengths given. Because BL is given as 6√5, and we have AB, BC, AC. Maybe we can compute the area via coordinates or using the formula involving the angle bisector.Alternatively, maybe using the formula for area in terms of the angle bisector.Wait, but I already found the sides via Stewart's theorem and the angle bisector theorem. Then using Heron's formula is a valid approach. However, getting such a messy number makes me wonder if there's a computational error. Let me verify the steps again.Starting from Stewart's theorem:AB² * LC + BC² * AL = AC * (BL² + AL * LC)We had AB = 12, BC = 16, AL = 3, LC = 4, BL = 6√5, AC = 7.Plugging in:12² * 4 + 16² * 3 = 7 * ( (6√5)^2 + 3*4 )Compute left-hand side:144 * 4 + 256 * 3 = 576 + 768 = 1344Right-hand side:7 * ( 180 + 12 ) = 7 * 192 = 1344So that checks out. So the sides are indeed 12, 16, 7.Then Heron's formula gives s = 35/2 = 17.5Area = sqrt[17.5 * 5.5 * 1.5 * 10.5]Wait, 17.5 * 10.5 = 183.75 and 5.5 * 1.5 = 8.25, then 183.75 * 8.25 = 1515.9375. Hmm, sqrt(1515.9375). Let me compute this.First, note that 38^2 = 1444, 39^2 = 1521. So sqrt(1515.9375) is between 38 and 39. Let's compute 38.9^2:38.9^2 = (38 + 0.9)^2 = 38^2 + 2*38*0.9 + 0.9^2 = 1444 + 68.4 + 0.81 = 1513.21Hmm, 38.9^2 = 1513.21, which is less than 1515.9375.38.95^2 = (38.9 + 0.05)^2 = 38.9^2 + 2*38.9*0.05 + 0.05^2 = 1513.21 + 3.89 + 0.0025 = 1517.1025That's more than 1515.9375. So sqrt(1515.9375) is approximately between 38.9 and 38.95. Let's try 38.93:38.93^2 = ?38.93^2: Let's compute 38^2 = 1444, 0.93^2 = 0.8649, and cross term 2*38*0.93 = 70.68. So total: 1444 + 70.68 + 0.8649 = 1515.5449Close to 1515.9375. So 38.93^2 ≈ 1515.5449, which is about 0.3926 less than 1515.9375. Then 38.94^2:38.94^2 = (38.93 + 0.01)^2 = 38.93^2 + 2*38.93*0.01 + 0.01^2 = 1515.5449 + 0.7786 + 0.0001 ≈ 1516.3236Wait, that seems off. Wait, actually, the difference between 38.93^2 and 38.94^2 is 0.7786 + 0.0001 + (1515.5449 to 1516.3236), which is a difference of about 0.7787. But we need to go from 1515.5449 to 1515.9375, which is a difference of 0.3926. So we can approximate:0.3926 / 0.7787 ≈ 0.504 of the interval between 38.93 and 38.94. So approximate sqrt(1515.9375) ≈ 38.93 + 0.504*0.01 ≈ 38.935So approximately 38.935. But this is a decimal approximation. However, the problem likely expects an exact value.Wait, but Heron's formula gave us the area as sqrt(1515.9375). Let me express 1515.9375 as a fraction.1515.9375 = 1515 + 15/16. Because 0.9375 = 15/16. So 1515.9375 = 1515 + 15/16 = (1515*16 + 15)/16 = (24240 + 15)/16 = 24255/16.Therefore, the area is sqrt(24255/16) = (sqrt(24255))/4.But can sqrt(24255) be simplified?Let's factor 24255.Divide by 5: 24255 ÷ 5 = 48514851 ÷ 3 = 16171617 ÷ 3 = 539539 ÷ 7 = 7777 ÷ 7 = 11So prime factors of 24255: 5 * 3 * 3 * 7 * 7 * 11Therefore, sqrt(24255) = sqrt(5 * 3² * 7² * 11) = 3 * 7 * sqrt(5 * 11) = 21 * sqrt(55)Therefore, the area is (21 * sqrt(55))/4So sqrt(24255)/4 = 21√55 / 4Wait, that seems a lot simpler. So the area is 21√55 divided by 4. Let me confirm that.Given that 24255 = 5 * 3² * 7² * 11, so sqrt(24255) = sqrt(3² * 7² * 5 * 11) = 3 * 7 * sqrt(55) = 21√55. Yes, correct.Therefore, the area is 21√55 / 4. So that would be the exact value. But let me check again.Alternatively, maybe there's a different approach that gives the area more straightforwardly. Let me think.Alternatively, since we know the lengths AL = 3, CL = 4, and BL = 6√5, maybe we can compute the area by dividing the triangle into two smaller triangles: ABL and BLC, each with base AL and CL respectively, and height from B. But since BL is an angle bisector, maybe we can use the formula for area in terms of angle bisectors.Alternatively, since we have the lengths of BL, AL, and CL, perhaps we can use the formula for the area based on the angle bisector length. Let me recall that.Alternatively, we can use coordinates. Let me try that approach. Place point A at (0, 0), point C at (7, 0), since AC is 7 units long. Then point L is at (3, 0), since AL = 3. Then point B is somewhere in the plane, and BL is the angle bisector with length 6√5. Let me set coordinates:Let’s set coordinate system such that point A is at (0,0), point C is at (7,0). Then point L is at (3,0). Let point B have coordinates (x,y). Then BL is the distance from B to L, which is sqrt((x - 3)^2 + y^2) = 6√5. So,(x - 3)^2 + y^2 = (6√5)^2 = 36 * 5 = 180.Also, from the Angle Bisector Theorem, we had AB = 12 and BC = 16. Wait, but in coordinate terms, AB is the distance from (0,0) to (x,y), so sqrt(x^2 + y^2) = 12, and BC is the distance from (7,0) to (x,y), so sqrt((x - 7)^2 + y^2) = 16.So now we have two equations:1. sqrt(x^2 + y^2) = 12 --> x^2 + y^2 = 1442. sqrt((x - 7)^2 + y^2) = 16 --> (x - 7)^2 + y^2 = 256Subtracting equation 1 from equation 2:(x - 7)^2 + y^2 - x^2 - y^2 = 256 - 144Expanding (x - 7)^2:x^2 - 14x + 49 + y^2 - x^2 - y^2 = 112Simplify:-14x + 49 = 112-14x = 112 - 49 = 63x = -63 / 14 = -4.5So x = -9/2.Now, substitute x = -9/2 into equation 1:(-9/2)^2 + y^2 = 14481/4 + y^2 = 144y^2 = 144 - 81/4 = (576/4 - 81/4) = 495/4Therefore, y = sqrt(495/4) = (sqrt(495))/2 = (sqrt(9*55))/2 = (3√55)/2So the coordinates of B are (-9/2, (3√55)/2)Now, to compute the area of triangle ABC, we can use the coordinates.Points A(0,0), B(-9/2, 3√55/2), C(7,0)The area can be calculated using the shoelace formula.Formula:Area = |(Ax(By - Cy) + Bx(Cy - Ay) + Cx(Ay - By))/2|Substituting:Ax = 0, Ay = 0Bx = -9/2, By = 3√55/2Cx = 7, Cy = 0Plug into formula:Area = |(0*(3√55/2 - 0) + (-9/2)*(0 - 0) + 7*(0 - 3√55/2))/2|Simplify each term:First term: 0Second term: (-9/2)*(0 - 0) = 0Third term: 7*(0 - 3√55/2) = 7*(-3√55/2) = -21√55/2So total inside the absolute value:|0 + 0 -21√55/2| / 2 = | -21√55/2 | /2 = (21√55/2)/2 = 21√55/4So the area is 21√55/4, which matches the Heron's formula result. Therefore, the area of triangle ABC is 21√55 over 4.But let me check if there's any other possible interpretation of the problem where BL is not the angle bisector but some other bisector. Wait, the problem says "BL is the bisector of triangle ABC". Typically, in a triangle, a bisector refers to an angle bisector. So I think that is correct. Also, given that AL and LC are given, it's consistent with the angle bisector theorem.Therefore, after careful calculation using both Stewart's theorem leading to Heron's formula and coordinate geometry, the area is 21√55 divided by 4. So that's the answer.Final AnswerThe area of triangle ( ABC ) is boxed{dfrac{21sqrt{55}}{4}}.</think>Given triangle (ABC) with (BL) as the angle bisector, we need to find its area given (|AL| = 3), (|BL| = 6sqrt{5}), and (|CL| = 4).1. Angle Bisector Theorem: The ratio of segments (AL) and (LC) is equal to the ratio of sides (AB) and (BC): [ frac{AL}{LC} = frac{AB}{BC} implies frac{3}{4} = frac{AB}{BC} implies AB = frac{3}{4}BC ]2. Stewart's Theorem: For cevian (BL) in triangle (ABC): [ AB^2 cdot LC + BC^2 cdot AL = AC cdot (BL^2 + AL cdot LC) ] Substituting the given values: [ AB^2 cdot 4 + BC^2 cdot 3 = 7 cdot ((6sqrt{5})^2 + 3 cdot 4) ] Simplifying the right-hand side: [ 4AB^2 + 3BC^2 = 7 cdot (180 + 12) = 7 cdot 192 = 1344 ]3. Substitute (AB = frac{3}{4}BC) into the equation: [ 4 left(frac{9}{16}BC^2right) + 3BC^2 = 1344 implies frac{9}{4}BC^2 + 3BC^2 = 1344 implies frac{21}{4}BC^2 = 1344 implies BC^2 = 256 implies BC = 16 ] Then, (AB = frac{3}{4} times 16 = 12).4. Heron's Formula: With sides (AB = 12), (BC = 16), and (AC = 7): [ s = frac{12 + 16 + 7}{2} = 17.5 ] The area is: [ sqrt{s(s - AB)(s - BC)(s - AC)} = sqrt{17.5 times 5.5 times 1.5 times 10.5} ] Converting to fractions and simplifying: [ 17.5 times 10.5 = frac{35}{2} times frac{21}{2} = frac{735}{4}, quad 5.5 times 1.5 = frac{11}{2} times frac{3}{2} = frac{33}{4} ] [ sqrt{frac{735}{4} times frac{33}{4}} = sqrt{frac{24255}{16}} = frac{sqrt{24255}}{4} = frac{21sqrt{55}}{4} ]5. Verification with Coordinates: Placing points (A(0,0)), (C(7,0)), and solving coordinates for (B) using distances (AB = 12) and (BC = 16), then using the shoelace formula confirms the area.Thus, the area of triangle (ABC) is (boxed{dfrac{21sqrt{55}}{4}}).]

🔑:Answer: frac{21 sqrt{55}}{4} approx 38.94.

🧠:[<think>Okay, let me try to figure out this problem step by step. So, the problem says that several identical books and identical albums were bought. The total cost for the books is 10 rubles and 56 kopecks. We need to find out how many books were bought. The conditions are that the price of one book is more than one ruble higher than the price of an album, and they bought 6 more books than albums. Hmm, alright, let's break this down.First, I need to remember that in Russian currency, 1 ruble is 100 kopecks. So, 10 rubles 56 kopecks is the same as 1056 kopecks. Maybe working in kopecks will make the calculations easier, so I'll convert everything to kopecks. That way, I don't have to deal with decimal points.Let me define some variables. Let’s say the number of albums bought is A. Then, since they bought 6 more books than albums, the number of books bought would be A + 6. Let’s denote the price of one album as P kopecks. Then, the price of one book would be P + 100 kopecks because it's more than one ruble higher. Wait, the problem says "more than one ruble higher," which is 100 kopecks. But does that mean at least 101 kopecks more? Hmm, the problem says "more than one ruble higher," so maybe strictly greater than 100 kopecks. So, the price difference is over 100 kopecks, so P_book = P_album + 101 kopecks or more. But maybe the price difference is an integer number of kopecks? Hmm, this could complicate things because we don't know the exact difference, just that it's more than one ruble. Wait, maybe we can denote the difference as D kopecks where D > 100. But then we have two variables: the price of the album and the price difference. But the problem might be solvable with the given information. Let's see.The total cost for the books is 1056 kopecks. The number of books is A + 6, each costing (P + D) kopecks, where D > 100. The total cost is then (A + 6)(P + D) = 1056. However, we also have that the albums cost some amount, but the problem doesn't mention the total cost for the albums. Wait, the problem only gives the total cost for the books. So maybe we don't need to consider the total cost of the albums? Hmm.Wait, let me read the problem again: "Several identical books and identical albums were bought. The books cost 10 rubles 56 kopecks. How many books were bought if the price of one book is more than one ruble higher than the price of an album, and 6 more books were bought than albums?" So, the total cost for all the books is 10 rubles 56 kopecks. We need to find the number of books. The constraints are that each book is more than one ruble (i.e., more than 100 kopecks) more expensive than an album, and the number of books is 6 more than the number of albums.So, perhaps we can model this with the number of albums as A, number of books as A + 6. Let the price of an album be Q kopecks, then the price of a book is Q + D kopecks where D > 100. The total cost for the books is (A + 6)(Q + D) = 1056 kopecks. However, we don't know Q or D. But maybe there's another way. Since we have two variables here: A and Q (and D, but D is related to Q if we consider that the book price is Q + D). Wait, but the problem doesn't give the total cost for the albums, so maybe we can't relate A and Q through that. Hmm. That complicates things because we have one equation and multiple variables.Wait, but perhaps the problem is set up such that the prices are integers in kopecks. So, Q and D are integers, and D > 100. Then, (A + 6)(Q + D) = 1056. We need to find integer A such that A + 6 divides 1056, and Q + D is 1056 / (A + 6). But also, since D > 100, then Q + D - Q = D > 100, so the price of a book is more than 100 kopecks higher than the album. So, book price - album price > 100 kopecks. So, (1056 / (A + 6)) - Q > 100. But we don't know Q. However, Q is the price of an album, which must be a positive integer. So, (1056 / (A + 6)) - Q > 100 => Q < (1056 / (A + 6)) - 100. Also, since Q must be positive, (1056 / (A + 6)) - 100 > 0 => 1056 / (A + 6) > 100 => A + 6 < 1056 / 100 => A + 6 < 10.56. Since A + 6 must be an integer, A + 6 <= 10. So, the number of books is at most 10. Wait, but if A + 6 <= 10, then A <= 4. But let's check.Wait, if (1056 / (A + 6)) > 100, then A + 6 < 1056 / 100 = 10.56, so A + 6 <= 10. Therefore, possible values for the number of books (A + 6) are integers from 1 to 10. But the number of books must be 6 more than the number of albums, so the number of books must be at least 7 (if A = 1). But since the total cost for books is 1056 kopecks, the number of books must divide 1056. Wait, 1056 divided by the number of books must be an integer because the price per book is in kopecks. So, the price per book is 1056 / (A + 6), which must be an integer. Therefore, (A + 6) must be a divisor of 1056. So, possible values for (A + 6) are the divisors of 1056 that are <=10 (since A + 6 <=10). Let's list the divisors of 1056 up to 10.First, factorize 1056:1056 ÷ 2 = 528528 ÷ 2 = 264264 ÷ 2 = 132132 ÷ 2 = 6666 ÷ 2 = 3333 ÷ 3 = 1111 is prime.So, prime factors: 2^5 * 3 * 11.Divisors up to 10:1, 2, 3, 4, 6, 8, (since 11 is over 10, 2^3=8 is the highest power of 2, and 3*2=6, etc.)Wait, let's list all the divisors of 1056:1, 2, 3, 4, 6, 8, 11, 12, 16, 22, 24, 32, 33, 44, 48, 66, 88, 96, 132, 176, 264, 352, 528, 1056But we need divisors less than or equal to 10. So, 1, 2, 3, 4, 6, 8.But the number of books is A + 6, which must be at least 7 (since A is the number of albums, which can't be negative; the smallest A can be is 0, making books 6, but then if A=0, albums bought would be 0, but the problem says "several" books and albums, so at least one of each? Maybe. Let's check the problem statement: "Several identical books and identical albums were bought." "Several" usually means more than one, but maybe it's just plural, so at least two? Or maybe at least one? If A=0, then number of books is 6. But buying 0 albums is allowed? The problem doesn't specify that they bought both books and albums, just "several identical books and identical albums were bought." Maybe "several" for each, so at least two of each? Hmm, but the Russian original might use a word that implies multiple. Anyway, perhaps we can consider the divisors of 1056 up to 10, which are 1,2,3,4,6,8. But if the number of books is A + 6, then possible number of books is one of these. However, if A + 6 is 1, then A = -5, which is impossible. Similarly, A + 6 = 2 would imply A = -4, which is invalid. Similarly, 3: A = -3; 4: A = -2; 6: A=0; 8: A=2. So, only when number of books is 8, A=2, or number of books is 6, A=0.But if A must be at least 1, then A=2 is the only possible, leading to number of books as 8. Let's check if 8 is a divisor of 1056. 1056 ÷ 8 = 132. So, 132 kopecks per book. Then, the price of the album would be 132 - D, where D > 100. So, the album price would be less than 132 - 100 = 32 kopecks. So, the album would cost 31 kopecks or less. But is that plausible? Let's see. If the album is 31 kopecks, then the book is 132 kopecks, which is 101 kopecks more. That satisfies the "more than one ruble higher" condition. However, the problem states that "the price of one book is more than one ruble higher than the price of an album," so the difference must be greater than 100 kopecks, which in this case is 132 - 31 = 101 kopecks, which is more than 100. So that's okay. But could there be another divisor?Wait, if we consider A + 6 = 8, then A = 2. So, they bought 2 albums and 8 books. The total cost for books is 8 * 132 = 1056 kopecks, which is correct. The price of the album would be 132 - D, but since D is the difference, which is 101 or more. Wait, but the difference is 132 - Q > 100, so Q < 132 - 100 = 32. So Q could be 31, 30, ..., 1. But if Q is 31, then the total cost for albums would be 2 * 31 = 62 kopecks. The problem doesn't mention the total cost for albums, so maybe that's acceptable. But maybe we need to check if the prices make sense. However, since the problem only gives the total cost for the books, maybe we don't need to consider the album's total cost.Alternatively, if the number of books is 6, then A = 0. But buying zero albums might not be allowed if "several" implies at least one. If allowed, then number of books is 6, each costing 1056 / 6 = 176 kopecks. Then, the price difference would be 176 - Q > 100 => Q < 76. So, Q could be 75 kopecks or less. But since they bought zero albums, the price of the album is irrelevant. But if the problem requires that they bought both books and albums, then A must be at least 1. So, A=2, number of books=8 is the only possible.Wait, but let's check if there are any other divisors. For example, 1056 ÷ 11 = 96, but 11 is greater than 10, which we previously considered as the upper limit for the number of books. But wait, earlier we thought that A + 6 must be <=10 because (1056 / (A + 6)) > 100. Wait, let me re-examine that step.We have that the price per book is 1056 / (A + 6). The price of the album is Q = (1056 / (A + 6)) - D, but D > 100. So, Q = price per book - D. Since Q must be positive, then price per book - D > 0 => D < price per book. But D > 100, so 100 < D < price per book. Therefore, price per book must be greater than 100. But since the price per book is 1056 / (A + 6), we have 1056 / (A + 6) > 100 => (A + 6) < 1056 / 100 = 10.56, so A + 6 <= 10. Therefore, number of books (A + 6) can be at most 10, so the divisors of 1056 up to 10 are 1, 2, 3, 4, 6, 8. As before.But if we take A + 6 = 8, then price per book is 132, as above. If we take A + 6 = 6, then price per book is 176, but that would require A = 0. If A + 6 = 4, price per book is 264, A = -2: invalid. Similarly for others. So, only possible valid number of books is 8 or 6. But 6 would require A=0, which might not be allowed. So, assuming they bought at least one album, then the number of books is 8.But let's check if there are other possible divisors. Wait, 1056 divided by 8 is 132, which gives a valid solution. If we consider A + 6 = 11, but 11 is a divisor of 1056 (1056 ÷ 11 = 96), but earlier we thought A + 6 must be <=10. Wait, why did we think that? Because if price per book is 1056 / (A + 6), and since price per book must be greater than 100 + Q, but Q is at least 1 kopeck, so price per book must be at least 101 kopecks? Wait, no, the price per book is more than one ruble higher than the album. So, if the album is Q kopecks, the book is Q + D where D > 100. Therefore, the price per book is Q + D, so the price per book must be at least Q + 101. But Q is at least 1 kopeck, so the minimum price per book is 1 + 101 = 102 kopecks. Therefore, price per book must be at least 102 kopecks, so 1056 / (A + 6) >= 102 => (A + 6) <= 1056 / 102 ≈ 10.35. So, A + 6 <=10. So, same conclusion.Therefore, possible number of books is 8 or 6. If 6, then A=0. But if A=0, then the price per book is 176, and the price of the album is 176 - D where D >100. So, the album price would be less than 76 kopecks. But since no albums were bought, the price of the album is irrelevant. However, the problem states that "several identical books and identical albums were bought." If "several" means at least one, then A cannot be 0, so the only possible solution is 8 books. But maybe the problem allows A=0. Let's check the original problem statement.Original problem: "Several identical books and identical albums were bought." The Russian word for "several" could be "несколько", which can mean "some" or "a few", but it doesn't necessarily specify a minimum number. However, it's plural, so "several books" would be at least two books, and "several albums" would be at least two albums. Wait, maybe. In Russian, "несколько книг и несколько альбомов" – несколько typically means "a few" or "several", which is more than one but not many. So, if it's plural, then "several" would mean at least two. Therefore, A >=2 and number of books = A + 6 >=8. So, if A >=2, then number of books >=8. Then, from the previous calculation, the only possible number of books is 8, since 8 is a divisor of 1056, leading to A=2. If number of books is 8, then price per book is 132. Then, the price of the album is 132 - D, where D >100. So, the album price would be less than 32 kopecks. For example, if D=101, then album price is 132 -101=31 kopecks. If D=102, album price is 30 kopecks, etc. But since they bought 2 albums, the total cost for albums is 2*31=62 kopecks or whatever, but since the problem doesn't mention the total cost for albums, this is acceptable. So, this seems to satisfy all conditions.Wait, but is there another divisor? Let's see. For example, if number of books is 12, which is a divisor of 1056, but 1056 /12=88. So, price per book would be 88 kopecks. Then, the price of the album is 88 - D, D>100. Then, 88 - D >0 => D <88. But D>100, which is impossible. So, 12 is invalid. Similarly, 16: 1056/16=66, then album price would be 66 - D, D>100 => 66 - D >0 => D <66, but D>100, impossible. So, those are invalid. So, only the divisors less than or equal to 10. So, 8 and 6.But if we assume that A >=2, then 8 books is the answer. But let's check if 6 books is possible. If number of books is 6, then price per book is 1056 /6=176 kopecks. Then, the price of the album is 176 - D, D>100. So, the album price is less than 176 -100=76 kopecks. If A=0, but as discussed, maybe A needs to be at least 2. If A=0 is allowed, then this is another solution. So, which one is correct?The problem says "several identical books and identical albums were bought." The word "several" can be ambiguous. In some contexts, "several" might mean "multiple" or "more than one," which would require at least two of each. But in Russian, "несколько" can sometimes be used for "some" as in "an unspecified number," which could include one. However, since it's plural, "книг" and "альбомов" are plural, which in Russian requires the number to be more than one. Wait, in Russian, the plural is used for quantities ending with 2,3,4, etc., but for numbers ending with 1 (except 11), it's singular. So, if they bought 1 album, it would be "альбом", but since the problem says "альбомов" (plural), that suggests that the number of albums is more than one. Similarly for books: "книг" is plural, so number of books is more than one. Therefore, A (number of albums) must be at least 2, leading to number of books A + 6 >=8. So, the only possible solution is 8 books.Therefore, the answer is 8.But let me verify once again. If number of books is 8, then price per book is 1056 /8=132 kopecks. The price of the album is 132 - D where D>100, so album price is less than 32. Since they bought A=2 albums, the total cost for albums is 2*(132 - D). But since D>100, 132 - D <32, so 2*(something less than 32) is less than 64 kopecks. But since the problem doesn't mention the total cost for albums, this is acceptable. Therefore, the answer is 8 books.Final AnswerThe number of books bought is boxed{8}.</think>Given that several identical books and identical albums were bought, with the total cost for the books being 10 rubles 56 kopecks (1056 kopecks), we need to determine how many books were bought. The price of one book is more than one ruble (100 kopecks) higher than the price of an album, and 6 more books were bought than albums.1. Convert the total cost for the books to kopecks: 10 rubles 56 kopecks = 1056 kopecks.2. Let ( A ) be the number of albums bought. Then, the number of books bought is ( A + 6 ).3. Let the price of one album be ( Q ) kopecks. The price of one book is ( Q + D ) kopecks, where ( D > 100 ).4. The total cost for the books is given by ((A + 6)(Q + D) = 1056).5. Since ( Q ) must be a positive integer, ( Q + D ) must be a divisor of 1056. Additionally, ( Q + D ) must be greater than 100 kopecks more than ( Q ), implying ( D > 100 ).We need to find the divisors of 1056 that are less than or equal to 10 (since ( A + 6 ) must be an integer and the price per book must be more than 100 kopecks higher than the album price). The valid divisors of 1056 up to 10 are 1, 2, 3, 4, 6, and 8. - If ( A + 6 = 8 ), then ( A = 2 ). The price per book is ( 1056 / 8 = 132 ) kopecks. The price difference ( D ) must be greater than 100, so ( Q = 132 - D ). Since ( D > 100 ), ( Q ) must be less than 32 kopecks, which is valid.- If ( A + 6 = 6 ), then ( A = 0 ). However, "several" implies at least one album, so ( A = 0 ) is invalid.Thus, the valid number of books bought is ( 8 ).[boxed{8}],<think>Okay, let's try to figure out this problem step by step. So, the problem is about buying some books and albums. They are identical, meaning each book has the same price and each album has the same price. The total cost for the books is 10 rubles and 56 kopecks. We need to find out how many books were bought. The clues given are that a book is more than one ruble expensive than an album, and they bought 6 more books than albums.First, let me recall that 1 ruble is 100 kopecks, so 10 rubles 56 kopecks is 1056 kopecks. That might be easier to work with since dealing with kopecks as whole numbers can avoid decimal issues.Let me define some variables. Let's let the number of albums be A. Then the number of books would be A + 6, since they bought 6 more books than albums.Let the price of one album be P kopecks. Then the price of one book would be P + 100 kopecks more than one ruble, right? Wait, the problem says "more than one ruble higher." So, the price of a book is more than 1 ruble higher than an album. Since 1 ruble is 100 kopecks, that means the price difference is more than 100 kopecks. So, if the album is P kopecks, the book is P + 101 kopecks or more. Hmm, but maybe we need to be precise here. The problem states "more than one ruble higher," which strictly means greater than 100 kopecks. So the difference is strictly greater than 100. So, book price = album price + at least 101 kopecks.But maybe the difference is an integer number of kopecks, since prices are in kopecks. So, the minimal possible difference is 101 kopecks.But perhaps we can think of the difference as some amount over 100 kopecks. Let's denote the price of an album as A_price kopecks, and the price of a book as B_price = A_price + D, where D > 100 kopecks.But we don't know D, so maybe we need to find both the price of the album, the price of the book, and the number of books and albums. Hmm, but the total cost for the books is 1056 kopecks. Wait, total cost for the books is given as 10 rubles 56 kopecks, which is 1056 kopecks. So, the total cost for books is number of books multiplied by price per book. Similarly, albums would have their own total cost, but the problem doesn't mention the total cost for albums. Wait, actually, the problem only mentions that the books cost 10 rubles 56 kopecks. It doesn't say anything about the total cost of albums. So maybe we don't need to consider the albums' total cost. Wait, let me check the problem again."Several identical books and identical albums were bought. The books cost 10 rubles 56 kopecks. How many books were bought if the price of one book is more than one ruble higher than the price of an album, and 6 more books were bought than albums?"So, the total amount spent on books is 1056 kopecks. We need to find the number of books. The price per book is more than 1 ruble (100 kopecks) higher than the price per album. Also, the number of books is 6 more than the number of albums.So, variables:Let A = number of albumsThen number of books = A + 6Let price per album = p kopecksThen price per book = p + d kopecks, where d > 100 kopecksTotal cost for books: (A + 6)(p + d) = 1056 kopecksBut we don't know A, p, or d. Hmm, this seems like we have multiple variables. But perhaps there are constraints that can help us.We need to find integer solutions since the number of books and albums, as well as their prices in kopecks, must be integers.So, this is a Diophantine equation problem. Let me try to structure it.Let me denote:Number of books = B = A + 6Price per book = q = p + d, where d > 100Total cost for books: B * q = 1056So, (A + 6)(p + d) = 1056But we also need to relate p and d. However, the problem only states that d > 100. So we don't have another equation, but maybe we can find possible factors of 1056 that fit the constraints.Since B and q must be positive integers, and B = A + 6, which is at least 6 (since A can be 0, but probably they bought at least one album? Wait, the problem says several books and albums were bought. So A and B are at least 1. Therefore, A >= 1, B >= 7.So, the number of books is B = A + 6, and B * q = 1056. So, B must be a divisor of 1056. Let's find all divisors of 1056 and see which ones could correspond to B >=7, and then check if the corresponding q = 1056 / B satisfies q - p > 100, where p is the price of an album. But we need to relate p to something. Wait, but we don't have the total cost for the albums, so how do we know p?Wait, perhaps the key is that the price of the album p must be a positive integer, and when you have the price of the book q = p + d, with d > 100, then p = q - d < q - 100. But we don't know p. However, since the total cost for the books is 1056, and the number of albums is B - 6, but we don't know the total cost for albums.Wait, maybe there's something missing here. How can we find both p and B with the given information? Let me think.We have:1. B = A + 62. Total book cost: B * q = 10563. q = p + d, where d > 100But without the total cost for albums or some other relation, how can we find p?Alternatively, maybe the price of the album is related to the price of the book through some other constraints. For example, maybe the price of the album is a factor that divides into some total, but we don't have the total. Wait, the problem doesn't mention the total spent on albums, only that books cost 1056 kopecks. So perhaps we can ignore the album cost? But then we have two variables: p and B. Let me see.Wait, B is the number of books, which is equal to A + 6, where A is the number of albums. But since we don't know A or B, and we don't know p (price of album) or q (price of book), except that q = p + d, d > 100, and B * q = 1056. So, we can model this as B * (p + d) = 1056, with d > 100, and B = A + 6, A = B - 6. But since A must be a positive integer, B must be at least 7.So, perhaps we can express p in terms of q and d, but since we don't know d, maybe another approach is needed.Alternatively, since q = p + d, and d > 100, then q > p + 100. Therefore, p < q - 100. Also, since the number of albums is A = B - 6, and the total cost for albums would be A * p. But we don't know that total. However, since A * p must also be an integer, but without knowing the total, it's hard to use that.Wait, maybe we can set up inequalities. Let's see:We have B = A + 6, so A = B - 6The price per book is q = p + d > p + 100Total cost for books: B * q = 1056So, q = 1056 / BTherefore, p = q - d < q - 100So, p < (1056 / B) - 100But we also know that the total cost for albums is A * p = (B - 6) * pBut since we don't know the total cost for albums, maybe we can't proceed this way.Wait, but perhaps we can use the fact that p must be a positive integer, so (1056 / B) - 100 > 0, because p must be positive.Therefore, (1056 / B) - 100 > 0=> 1056 / B > 100=> 1056 > 100 B=> B < 1056 / 100=> B < 10.56Since B is an integer, B <= 10But we also know that B >=7 (since A = B -6 >=1)Therefore, possible values of B are 7, 8, 9, 10So, B can be 7, 8, 9, 10Then, let's check these possible values:For each B in 7,8,9,10:Calculate q = 1056 / BCheck if q is an integer (since price per book must be integer kopecks)Also, check if p = q - d (where d >100) and p must be a positive integer. But since d is also an integer, p = q - d < q -100. Therefore, p <= q -101 (since d must be at least 101)So, for each B, compute q = 1056 / B. If q is integer, then check if p = (1056 / B) - 101 is positive. Because p must be positive, so (1056 / B) -101 >0So, let's check each B:B=7:q = 1056 /7 ≈ 150.857… Not integer. So discard.B=8:q=1056/8=132. So, q=132 kopecks. Then p must be less than 132 -100=32. Since p must be positive integer, p<=31. Also, since number of albums A=8-6=2. So, total cost for albums would be 2*p. But since we don't know total cost for albums, but p must be such that 2*p is some integer, but we don't have info. However, since we just need p to exist as a positive integer less than 32. So, possible? Yes. But is there any other constraints? Let me think. Wait, the problem states that "the price of one book is more than one ruble higher than the price of an album". So, the difference between book price and album price is more than 100 kopecks. So, q - p >100. Therefore, p < q -100. Since q=132, p < 32. So p can be any integer from 1 to 31. So, possible. But since we need to find the number of books, which is B=8. But we need to check if there's a possible p. However, since p can be any value from 1 to 31, there are multiple possibilities. So maybe B=8 is a possible answer? But we need to check other B's.Wait, but perhaps there are multiple possible B's. Let's check.B=9:1056 /9= 117.333… Not integer. Disregard.B=10:1056 /10=105.6. Not integer. Disregard.So the only B that gives an integer q is B=8, with q=132 kopecks. Then, since the price of the album p must satisfy p <132 -100=32, and p must be a positive integer. So p can be 1 to 31. But since the problem doesn't give more constraints, but asks "how many books were bought". If B=8 is possible, then maybe that's the answer. But wait, maybe we need to check if there is a corresponding integer p such that when you multiply by the number of albums (A=2), but since we don't have the total cost for the albums, maybe there is no constraint on p. So as long as p is a positive integer less than 32, which it is, then it's acceptable. Therefore, the answer is 8 books.Wait, but let me check again. If B=8, then A=2. The price per book is 132 kopecks. Then, the price per album is p, which is less than 32. So, p can be anything from 1 to 31. However, the problem says that "the price of one book is more than one ruble higher than the price of an album". So, 132 - p >100 => p <32. Which is already considered. So as long as p is less than 32, which is true, then the condition is satisfied. So, even though we don't know the exact value of p, the problem doesn't require us to find p, only the number of books. So, since B=8 is the only possible number that divides 1056 into an integer price per book with B >=7 and B<=10, then the answer is 8.But wait, let me check if there are other divisors of 1056 beyond 7-10. Because maybe B can be larger than 10. Wait, earlier I thought that since B <10.56, so B<=10, but actually, the calculation was (1056 / B) >100 => B <1056 /100=10.56. Therefore, B must be less than 10.56, so integer B<=10.But perhaps there are larger B's where q=1056/B is integer, but q - p >100. Wait, but if B is larger, then q is smaller. For example, if B=12, then q=1056/12=88. Then, p must be less than 88 -100= -12, which is impossible. So, no. Similarly, B=11: 1056/11=96. p <96 -100= -4, impossible. So, only B=8 is possible.Wait, but let's check B=6. But B must be at least 7 because A=B-6 must be at least 1. If B=6, then A=0, which is invalid. So yes, B must be from 7 to10.But only B=8 gives an integer q. Therefore, the answer is 8.But let me confirm this again. If 8 books were bought, each costing 132 kopecks, totaling 8*132=1056 kopecks. The number of albums is 8-6=2. The price of each album is less than 132-100=32 kopecks. So, for example, if the albums cost 31 kopecks each, then total album cost is 2*31=62 kopecks. But since the problem doesn't mention the total cost for albums, it's acceptable. The only constraints are met: 6 more books than albums, book price more than 1 ruble higher than album price. So yes, this works. Therefore, the answer is 8.Final AnswerThe number of books bought is boxed{8}.</think>Given that several identical books and albums were bought, where the total cost for the books is 10 rubles 56 kopecks (1056 kopecks), we need to determine the number of books bought. The price of one book is more than one ruble (100 kopecks) higher than the price of an album, and 6 more books were bought than albums.1. Define variables: - Let ( A ) be the number of albums. - The number of books is ( B = A + 6 ). - Let ( p ) be the price of an album in kopecks. - The price of a book is ( q = p + d ) where ( d > 100 ) kopecks.2. Total cost equation: - The total cost for the books is ( B times q = 1056 ) kopecks.3. Determine possible values for ( B ): - Since ( B times q = 1056 ), ( B ) must be a divisor of 1056. - Given ( B = A + 6 ) and ( A geq 1 ), ( B ) must be at least 7. - Additionally, ( q > p + 100 ) implies ( q - p > 100 ).4. Check possible divisors of 1056: - Possible values for ( B ) (considering ( 7 leq B leq 10 )): - ( B = 7 ): ( 1056 div 7 ) is not an integer. - ( B = 8 ): ( 1056 div 8 = 132 ) (integer). - ( B = 9 ): ( 1056 div 9 ) is not an integer. - ( B = 10 ): ( 1056 div 10 ) is not an integer.5. Verify constraints for ( B = 8 ): - The price per book ( q = 132 ) kopecks. - The number of albums ( A = B - 6 = 2 ). - The price of an album ( p ) must be less than ( 132 - 100 = 32 ) kopecks, which is feasible since ( p ) must be a positive integer.Thus, the number of books bought is (boxed{8}).]

🔑:5. Since each book is more expensive than a ruble, no more than 10 books were bought. Moreover, it is clear that no fewer than 7 books were bought (since at least one album was bought). The number 1056 is divisible by 8 and not divisible by 7,9,10. Therefore, 8 books were bought. (MSU, Mechanics and Mathematics, 1968)

🧠:[<think>Okay, so I need to find the equation of the line that's symmetric to the line y - 2x - 3 = 0 with respect to the line y = x + 1. Hmm, symmetry with respect to a line... That should mean that if I reflect the original line over the line y = x + 1, I get the symmetric line. But how do I actually do that?First, I remember that reflecting a line over another line involves some geometric transformations. Maybe I need to find the reflection of a couple of points on the original line and then determine the equation of the reflected line using those points? Or is there a more straightforward algebraic method?Let me recall. The general approach for finding the reflection of a line over another line is:1. Find the reflection of a point on the original line over the mirror line.2. Find the reflection of another point on the original line over the same mirror line.3. Then use those two reflected points to determine the equation of the reflected line.Alternatively, there might be a formula for the reflection of a line over another line, but I don't remember it off the top of my head. Maybe it's better to go with the step-by-step method.So let's try that.First, the original line is y - 2x - 3 = 0, which can be rewritten as y = 2x + 3. This is a straight line with slope 2 and y-intercept 3.The mirror line is y = x + 1, which is a straight line with slope 1 and y-intercept 1.To find the reflection of the line y = 2x + 3 over the line y = x + 1, I need to reflect points on the original line and then find the equation of the line through those reflected points.Let me choose two points on the original line. Let's pick two easy points.First, when x = 0, y = 2*0 + 3 = 3. So one point is (0, 3).Second, when x = 1, y = 2*1 + 3 = 5. So another point is (1, 5).Wait, but maybe even simpler. Let me check if these points are good. Let's verify (0,3) is on y = 2x + 3: yes. (1,5) is also on the line. Now, I need to reflect these two points over the line y = x + 1.So, first, let's recall how to reflect a point over a line. The formula for reflecting a point (a, b) over the line Ax + By + C = 0 is:The reflected point (a', b') is given by:a' = a - 2A(Aa + Bb + C)/(A² + B²)b' = b - 2B(Aa + Bb + C)/(A² + B²)But maybe I need to write the mirror line in standard form first. The mirror line is y = x + 1. Let's rearrange this to standard form:x - y + 1 = 0, so A = 1, B = -1, C = 1.So, the formula for reflecting a point (a, b) over the line x - y + 1 = 0 would be:a' = a - 2*1*(1*a + (-1)*b + 1)/(1² + (-1)²) = a - 2*(a - b + 1)/2 = a - (a - b + 1) = a - a + b - 1 = b - 1Similarly,b' = b - 2*(-1)*(1*a + (-1)*b + 1)/(1² + (-1)²) = b + 2*(a - b +1)/2 = b + (a - b +1) = a +1So, summarizing, the reflection of a point (a, b) over the line y = x + 1 is (b - 1, a + 1). Wait, that seems interesting. Let me verify this formula.Suppose we have a point (a, b). The reflection over the line y = x + 1.Alternatively, since the line y = x + 1 is a 45-degree line shifted up by 1, the reflection might involve swapping coordinates and adjusting for the shift.Alternatively, another way to think about reflecting over the line y = x + 1 is to perform a translation that maps the line y = x + 1 to y = x, then reflect over y = x, and then translate back.Let me try that approach.First, translate the coordinate system down by 1 unit. That is, let Y = y - 1. Then the line y = x + 1 becomes Y = x. Then, reflecting over Y = x is just swapping x and Y. Then, translating back by adding 1 to Y.So, let's formalize this.Given a point (x, y), translate down by 1: (x, Y) where Y = y - 1.Reflect over Y = x: swap x and Y, giving (Y, x).Translate back up by 1: (Y, x + 1).So, the reflection of (x, y) over y = x + 1 is (Y, x + 1) = (y - 1, x + 1).So, yes, the reflection formula is (x', y') = (y - 1, x + 1). So, this matches the earlier result.Therefore, the reflection of (a, b) over y = x +1 is (b -1, a +1).Therefore, to reflect a point (a, b), swap the coordinates and subtract 1 from the new x-coordinate and add 1 to the new y-coordinate.So, for our two points:First point: (0, 3). Reflecting gives (3 - 1, 0 + 1) = (2, 1).Second point: (1, 5). Reflecting gives (5 - 1, 1 + 1) = (4, 2).Wait, but hold on. Let me verify.Wait, according to the formula, reflection of (a, b) is (b -1, a +1). So, (0,3):x’ = 3 -1 = 2, y’ = 0 +1 =1. So (2,1). Correct.Similarly, (1,5):x’ =5 -1=4, y’=1 +1=2. So (4,2). Correct.So these are two points on the reflected line. Now, let's find the equation of the line passing through (2,1) and (4,2).First, compute the slope: m = (2 -1)/(4 -2)=1/2.So the slope is 1/2. Then, using point-slope form:Using point (2,1):y -1 = (1/2)(x -2)So y = (1/2)x -1 +1 = (1/2)x.Wait, that's interesting. So the reflected line is y = (1/2)x.But let's verify with the other point (4,2):y = (1/2)(4) = 2. Correct. So yes, the line passes through both points.Therefore, the equation of the reflected line is y = (1/2)x, or equivalently, 2y -x =0.But let me check if this is correct. Alternatively, maybe I should reflect another point to see.Take another point on the original line. Let's take x=2. Then y=2*2 +3=7. So point (2,7). Reflect over y=x+1:x’=7 -1=6, y’=2 +1=3. So (6,3).Check if (6,3) is on the line y=(1/2)x:3 = (1/2)*6=3. Correct.So yes, (6,3) is on the line. So that seems to check out.Therefore, the reflected line is y = (1/2)x, or x -2y =0.But wait, the problem says "Find the equation of the line that is symmetric to the line y−2x−3=0 with respect to the line y=x+1."So the original line is y -2x -3=0, which is y=2x+3. The symmetric line is y=(1/2)x, which can be written as y - (1/2)x =0, or multiplying both sides by 2: 2y -x =0. So x -2y =0. So the equation is x -2y=0.But let me check again with another approach to make sure.Alternatively, maybe we can use the formula for the reflection of a line over another line. Let me recall.If we have a line L1: ax + by +c =0, and we want to reflect it over the line L2: dx + ey + f =0, then the reflected line can be found by reflecting each point on L1 over L2, but that's essentially what we did before.Alternatively, maybe there's a way using angles or distance formulas.Alternatively, since reflection over a line preserves angles, so the angle between the original line and the mirror line should be equal to the angle between the reflected line and the mirror line.So, let's compute the angle between the original line and y=x+1, then compute the angle for the reflected line accordingly.The original line has slope 2, the mirror line has slope 1. The angle between them is arctan(2) - arctan(1) = arctan(2) - 45 degrees.The tangent of the angle between them is |(m2 - m1)/(1 + m1m2)| = |(1 -2)/(1 + 2*1)| = |(-1)/3|=1/3.Similarly, the reflected line should make the same angle with the mirror line. Let the slope of the reflected line be m. Then the tangent of the angle between m and 1 should also be 1/3.So |(m -1)/(1 + m*1)| =1/3.Therefore, (m -1)/(1 + m)= ±1/3.So solving for m:Case 1: (m -1)/(1 + m)=1/3Multiply both sides by (1 + m):m -1 = (1 + m)/3Multiply both sides by 3:3m -3 =1 + m3m - m =1 +32m=4 => m=2Case 2: (m -1)/(1 +m)= -1/3Multiply both sides by (1 +m):m -1= - (1 +m)/3Multiply both sides by 3:3m -3= -1 -m3m +m= -1 +34m=2 => m=0.5But the original line had slope 2, so the reflected line should not have slope 2 again. Therefore, the slope of the reflected line should be 0.5. Which matches our previous result. Therefore, the reflected line has slope 1/2.Therefore, combining this with the points we found earlier, the equation is y=(1/2)x.Therefore, the answer is y=(1/2)x, or x -2y=0.But let me check once more. Suppose we take the mirror line y=x+1, and the original line y=2x+3. Their intersection point is important because the reflected line should also pass through the reflection of this intersection point.Wait, actually, the intersection point of the original line and the mirror line will lie on both lines, so its reflection is itself. Therefore, the reflected line must pass through the same intersection point.Wait, is that true? Let's check.Find the intersection point of y=2x +3 and y=x +1.Set 2x +3 =x +1.2x -x =1 -3x= -2Then y= -2 +1= -1.So the intersection point is (-2, -1). Since this point is on both lines, reflecting it over the mirror line should give the same point. Therefore, the reflected line must pass through (-2, -1).But according to our previous reflected line y=(1/2)x, plugging x=-2, y=(1/2)*(-2)= -1. So y=-1. So the point (-2, -1) is indeed on the reflected line. Therefore, that checks out.Therefore, the reflected line passes through (-2, -1) and has slope 1/2. So that's consistent.Alternatively, we could have found one point (the intersection point) and the slope to determine the equation.But in any case, all these checks seem to confirm that the reflected line is y=(1/2)x.But wait, the original line is y -2x -3=0. The reflected line is y - (1/2)x=0. Let me check if this is indeed the reflection.Alternatively, maybe there's an algebraic method.Suppose that for any point (x, y) on the original line, its reflection (x', y') over y=x+1 lies on the reflected line. So, if we can express (x, y) in terms of (x', y'), and substitute into the original equation, we can find the equation of the reflected line.Given that reflection over y=x+1 maps (x, y) to (y -1, x +1). Therefore, the inverse reflection would map (x', y') to (y' -1, x' +1). Wait, actually, reflecting twice over the same line brings you back to the original point. So, if reflecting (x, y) gives (x', y'), then reflecting (x', y') should give back (x, y). Therefore, using the formula:If reflection of (a, b) is (b -1, a +1), then reflecting (b -1, a +1) should give ( (a +1) -1, (b -1) +1 ) = (a, b). Correct. So the inverse reflection is indeed (x', y') maps to (y' -1, x' +1).Therefore, if (x', y') is on the reflected line, then the original point (x, y)= (y' -1, x' +1) must lie on the original line y -2x -3=0.Therefore, substituting:(y' -1) -2(x' +1) -3=0Simplify:y' -1 -2x' -2 -3=0y' -2x' -6=0Therefore, y' -2x' -6=0 is the equation that (x', y') must satisfy. But this is the equation of the reflected line. Therefore, the reflected line is y -2x -6=0.Wait, hold on. Wait, that's conflicting with our previous result. Wait, this suggests that the reflected line is y -2x -6=0, which is y=2x +6. But earlier, reflecting points gave us y=(1/2)x. There's a discrepancy here. Which one is correct?Wait, this is confusing. Let me check.So according to the algebraic method: For any reflected point (x', y') on the reflected line, the original point (x, y) = (y' -1, x' +1) must lie on the original line y -2x -3=0.Therefore, substitute (y' -1) for x, and (x' +1) for y into the original equation:(x' +1) -2(y' -1) -3 =0Simplify:x' +1 -2y' +2 -3 =0x' -2y' + (1 +2 -3)=0x' -2y' +0=0Therefore, x' -2y' =0.Therefore, the equation of the reflected line is x -2y=0, which is the same as y=(1/2)x. So that matches our previous result. Wait, but in the previous step, I had substituted incorrectly. Let me check again.Wait, no. Wait, in the substitution step, the original line is y -2x -3=0. The original point is (x, y) = (y' -1, x' +1). Therefore, substituting into the original equation:y -2x -3 = (x' +1) -2(y' -1) -3 = x' +1 -2y' +2 -3 = x' -2y' +0 =0Therefore, x' -2y' =0. Therefore, the reflected line is x -2y =0. Therefore, this confirms the previous result. So why did I get confused earlier? Because I might have mixed up the variables.So, substituting (x, y) in terms of (x', y') gives the equation x' -2y'=0. Therefore, renaming (x', y') back to (x, y), the equation is x -2y=0, which is the same as y=(1/2)x. Therefore, this is consistent with the earlier method.Therefore, the correct equation is x -2y=0 or y=(1/2)x.Therefore, the answer is x -2y=0.But let me confirm once again by using another approach.Alternatively, the reflection of a line over another line can be found by using the formula for the reflection of a line. If we have a line L1 and a mirror line L2, then the reflection of L1 over L2 can be found by the following steps:1. Find the equations of L1 and L2.2. Find the point of intersection between L1 and L2.3. Find a direction vector of L1.4. Find the reflection of the direction vector over L2.5. Use the intersection point and the reflected direction vector to form the reflected line.Let's try this method.First, L1: y=2x +3, L2: y=x +1.Intersection point: As before, solving 2x +3 =x +1 gives x=-2, y=-1. So the point is (-2, -1).Direction vector of L1: Since L1 has slope 2, a direction vector is (1, 2).We need to reflect this direction vector over L2.To reflect a vector over a line, we can use vector projection.First, the direction vector of L2 is (1,1) since it has slope 1.The reflection of vector v over line with direction u is given by 2 proj_u v - v.But here, we need to reflect the direction vector (1,2) over the line L2, whose direction is (1,1).Wait, but actually, the reflection of a vector over another line involves the normal vector. Wait, perhaps I need to use the formula for reflecting a vector over a line in 2D.Alternatively, since the line L2 is y =x +1, which has a direction vector (1,1), and a normal vector (1,-1).Wait, let me recall that to reflect a vector over a line, you can decompose the vector into components parallel and perpendicular to the line, then reverse the perpendicular component.So, let's take the direction vector of L1: v = (1, 2).We need to find the reflection of v over L2. The line L2 has direction vector u = (1,1), and normal vector n = (1, -1).First, decompose v into parallel and perpendicular components.The parallel component is proj_u v = ( (v ⋅ u)/||u||² ) uCompute v ⋅ u =1*1 +2*1=3||u||²=1² +1²=2So proj_u v = (3/2)(1,1) = (3/2, 3/2)The perpendicular component is v - proj_u v = (1,2) - (3/2, 3/2) = (-1/2, 1/2)The reflection of v over L2 is proj_u v - perpendicular component = (3/2, 3/2) - (-1/2, 1/2) = (3/2 +1/2, 3/2 -1/2)= (2, 1)Therefore, the reflected direction vector is (2,1). Wait, but (2,1) as a direction vector has a slope of 1/2, which matches our previous result.Therefore, the reflected line passes through the intersection point (-2, -1) and has direction vector (2,1). Therefore, the slope is 1/2, as before. Therefore, the equation is y = (1/2)x + b. Plugging in (-2, -1):-1 = (1/2)*(-2) + b => -1 = -1 + b => b=0. Therefore, y=(1/2)x. Correct.Therefore, this method also leads us to the same conclusion.Therefore, regardless of the method used, the reflected line is y=(1/2)x, or x -2y=0.Therefore, the answer should be x -2y=0.But let me check once more with the original reflection of points.We had the original points (0,3), (1,5), (2,7) reflected to (2,1), (4,2), (6,3). All lying on y=(1/2)x. So yes, that seems correct.But let me check another point. Take x= -1 on the original line: y=2*(-1) +3=1. So point (-1,1). Reflect over y=x+1: x’=1 -1=0, y’=-1 +1=0. So (0,0). Check if (0,0) is on y=(1/2)x: 0=(1/2)*0=0. Yes. Correct.Therefore, all these verifications confirm that the reflected line is indeed y=(1/2)x.Therefore, the equation is y=(1/2)x, which can be written as x -2y=0.Hence, the answer is x -2y=0.Final AnswerThe equation of the symmetric line is boxed{x - 2y = 0}.</think>To find the equation of the line symmetric to the line ( y - 2x - 3 = 0 ) with respect to the line ( y = x + 1 ), we follow these steps:1. Reflect Points on the Original Line: - The original line ( y - 2x - 3 = 0 ) can be rewritten as ( y = 2x + 3 ). - Choose points on the original line, such as ((0, 3)) and ((1, 5)).2. Reflect These Points Over the Mirror Line ( y = x + 1 ): - The reflection formula for a point ((a, b)) over the line ( y = x + 1 ) is ((b - 1, a + 1)). - Reflecting ((0, 3)) gives ((3 - 1, 0 + 1) = (2, 1)). - Reflecting ((1, 5)) gives ((5 - 1, 1 + 1) = (4, 2)).3. Determine the Equation of the Reflected Line: - The reflected points ((2, 1)) and ((4, 2)) lie on the reflected line. - Calculate the slope ( m ) of the line through these points: ( m = frac{2 - 1}{4 - 2} = frac{1}{2} ). - Using the point-slope form with point ((2, 1)): ( y - 1 = frac{1}{2}(x - 2) ), which simplifies to ( y = frac{1}{2}x ).4. Verification: - Check if the intersection point of the original line and the mirror line lies on the reflected line. The intersection point ((-2, -1)) is on both lines. - Substitute ((-2, -1)) into the reflected line equation ( y = frac{1}{2}x ): ( -1 = frac{1}{2}(-2) ), which is true.5. Algebraic Method: - For any point ((x', y')) on the reflected line, the original point ((y' - 1, x' + 1)) must lie on the original line. Substituting ((y' - 1, x' + 1)) into ( y - 2x - 3 = 0 ) confirms the reflected line equation ( x - 2y = 0 ).Thus, the equation of the symmetric line is (boxed{x - 2y = 0}).,<think>Okay, so I need to find the equation of the line that's symmetric to the line y - 2x - 3 = 0 with respect to the line y = x + 1. Hmm, symmetric with respect to another line. I remember that reflecting a line over another line involves some geometry and maybe some coordinate transformations. Let me try to recall.First, maybe I should write down the given lines. The original line is y - 2x - 3 = 0, which can be rewritten as y = 2x + 3. The line of symmetry is y = x + 1. So, I need to find the reflection of y = 2x + 3 over the line y = x + 1.I think the general method for reflecting a line over another line involves finding the image of each point on the original line after reflection over the mirror line. But doing that for every point would be tedious. Maybe there's a smarter way.Alternatively, I remember that reflecting a line over another line can be done by reflecting two points on the original line over the mirror line and then finding the equation of the line through the two reflected points. That might work. Let me try that.So, pick two points on the original line y = 2x + 3. Let's choose two easy points. For example, when x = 0, y = 2*0 + 3 = 3. So point A is (0, 3). Another point, say when x = 1, y = 2*1 + 3 = 5. So point B is (1, 5). Now, I need to find the reflection of these two points over the line y = x + 1.Reflecting a point over a line. I recall the formula for reflecting a point (x, y) over the line ax + by + c = 0 is:The reflected point (x', y') is given by:x' = x - 2a(ax + by + c)/(a² + b²)y' = y - 2b(ax + by + c)/(a² + b²)But first, I need to write the mirror line y = x + 1 in the general form ax + by + c = 0. Let's rearrange that: x - y + 1 = 0. So, a = 1, b = -1, c = 1.So applying the reflection formula for a point (x, y):x' = x - 2*1*(1*x + (-1)*y + 1)/(1² + (-1)²) = x - 2*(x - y + 1)/2 = x - (x - y + 1) = x - x + y - 1 = y - 1Similarly,y' = y - 2*(-1)*(x - y + 1)/(1 + 1) = y + 2*(x - y + 1)/2 = y + (x - y + 1) = x + 1So the reflection of a point (x, y) over the line y = x + 1 is (y - 1, x + 1). Let me verify this formula with a simple example. Suppose we take the point (0, 1), which is on the mirror line y = x + 1 (since 1 = 0 + 1). Reflecting it should give the same point. Plugging into the formula: (1 - 1, 0 + 1) = (0, 1). Correct. Another point not on the line: say (0, 0). Reflecting over y = x +1. The formula gives (0 -1, 0 +1) = (-1, 1). Let's check if this is the reflection. The midpoint between (-1,1) and (0,0) should lie on the mirror line. Midpoint is (-0.5, 0.5). Does this satisfy y = x +1? 0.5 vs -0.5 +1 = 0.5. Yes. And the line connecting (0,0) and (-1,1) should be perpendicular to the mirror line. The slope of the mirror line is 1. The slope between (0,0) and (-1,1) is (1-0)/(-1-0) = -1, which is perpendicular. So the formula seems correct.Great, so now I can use this formula to reflect points A(0,3) and B(1,5).First, reflecting point A(0,3):x' = 3 - 1 = 2y' = 0 + 1 = 1So image of A is (2, 1).Reflecting point B(1,5):x' = 5 - 1 = 4y' = 1 + 1 = 2So image of B is (4, 2).Now, we need to find the equation of the line passing through the points (2,1) and (4,2).First, compute the slope:m = (2 - 1)/(4 - 2) = 1/2.Then, using point-slope form with point (2,1):y - 1 = (1/2)(x - 2)Multiply both sides by 2:2(y - 1) = x - 22y - 2 = x - 2Bring all terms to left:x - 2y = 0Or, rearranged: x - 2y = 0. Alternatively, y = (1/2)x.Wait, but let's check if this line is indeed the reflection.Alternatively, maybe I made a mistake in the calculation. Let's verify.First, the original line is y = 2x + 3. The mirror line is y = x +1. The reflected line is supposed to be x - 2y =0? Let's see.Alternatively, perhaps another approach would help. Let me think.Another method: the reflection of a line over another line can be found by first translating the coordinate system so that the mirror line passes through the origin, perform the reflection, and then translate back.Since the mirror line is y = x +1, which is a diagonal line shifted up. If we shift the coordinate system down by 1 unit, the mirror line becomes y = x. Then reflecting over y = x is just swapping x and y. Then shift back up.Let me try this approach.First, let's perform a translation. Let u = x, v = y -1. Then in terms of u and v, the mirror line y = x +1 becomes v +1 = u +1, so v = u. So the mirror line is v = u in the translated coordinates.Original line: y = 2x + 3. In terms of v: v +1 = 2u +3 => v = 2u + 2.Reflecting the line v = 2u + 2 over the line v = u. Reflecting over v = u is equivalent to swapping u and v. So the reflected line would be u = 2v + 2. Wait, no. Wait, reflecting a line over v = u is replacing u with v and v with u. So substituting u = 2v +2. Wait, no. Let me think. The original line is v = 2u +2. To find the reflection over v = u, which is the line of reflection, the image of a point (u, v) is (v, u). So the image of the line v = 2u +2 would be u = 2v +2. Solving for v: u - 2 = 2v => v = (u -2)/2.So the reflected line is v = (u -2)/2.Now, translating back to original coordinates. Remember u = x, v = y -1. So:y -1 = (x -2)/2Multiply both sides by 2:2(y -1) = x -22y -2 = x -2Bring all terms to left:x - 2y = 0Which is the same as before: x - 2y =0 or y = (1/2)x. So that's consistent.Therefore, the equation of the reflected line is x - 2y =0. Alternatively, y = (1/2)x.Wait, but let's check with the reflected points. Earlier, reflecting points A(0,3) and B(1,5) gave (2,1) and (4,2). The line through (2,1) and (4,2) is indeed y = (1/2)x. Plugging in x=2: y=1. Correct. x=4: y=2. Correct.So, the equation is y = (1/2)x or x - 2y =0. So, is this the correct answer? Let me check another way.Alternatively, maybe using the formula for reflecting a line over another line. Suppose we have the original line and the mirror line. The formula for the reflected line can be found using the formula here.But I might need to recall the formula. Alternatively, perhaps another approach is better.Suppose that the original line is y = 2x + 3. The mirror line is y = x +1. Let me find the angle between the original line and the mirror line, then construct the reflected line with the same angle on the other side.But maybe that's more complicated. Alternatively, to find where the original line and the mirror line intersect, then the reflected line must pass through the reflection of that intersection point. But since the intersection point is on the mirror line, its reflection is itself.So first, find the intersection point of the original line and the mirror line.Original line: y = 2x + 3.Mirror line: y = x +1.Set equal: 2x +3 = x +1 => x = -2. Then y = (-2) +1 = -1. So intersection point is (-2, -1).Since this point is on the mirror line, its reflection is itself. Therefore, the reflected line must pass through (-2, -1).Additionally, the reflected line should have a slope that is the reflection of the original slope over the mirror line's slope.The original line has slope 2. The mirror line has slope 1. The angle between them can be calculated, but perhaps it's easier to compute the slope of the reflected line using some formula.I recall that if a line with slope m is reflected over a line with slope k, then the slope m' of the reflected line is given by:m' = ( (1 - k m) + k (1 + m k) ) / ( (1 + k m) - k (1 - m k) )Wait, maybe that's not the right formula. Let me recall the formula for the slope of the reflected line.Alternatively, the formula for the tangent of the angle between two lines. Suppose the original line makes an angle θ with the mirror line, then the reflected line will make an angle θ on the other side.Alternatively, the formula for the slope of the reflected line can be derived using the tangent of twice the angle between the mirror line and the original line.Wait, let's consider the angle between the original line and the mirror line. Let me compute the angle θ between them.The slope of the mirror line is 1, so angle α = 45 degrees. The original line has slope 2, so angle β = arctan(2). The angle θ between them is β - α.The reflected line will have an angle α - θ on the other side. Wait, no. If the original line makes an angle θ with the mirror line, then the reflected line makes the same angle θ on the other side. So the angle of the reflected line with respect to the mirror line is θ, but in the opposite direction.Alternatively, using the formula for reflection of a line over another line. Let me check.Alternatively, since we know that the reflection of a line over another line can be found by reflecting two points on the original line and then finding the line through those reflected points. Which we already did, and got the line x - 2y =0. But let's verify this with another method.Alternatively, suppose we take a general point (x, y) on the original line y = 2x + 3. Its reflection over the mirror line y = x +1 is (y -1, x +1). So the reflection of (x, 2x +3) is (2x +3 -1, x +1) = (2x +2, x +1). Let this image point be (X, Y). So X = 2x +2 and Y = x +1. Express x in terms of X and Y. From Y = x +1, x = Y -1. Substitute into X = 2x +2: X = 2(Y -1) +2 = 2Y -2 +2 = 2Y. Therefore, X = 2Y, which is the equation Y = (1/2)X. So in terms of original coordinates, the reflected line is y = (1/2)x. Therefore, the equation is x - 2y =0. So this matches our previous result.Therefore, the reflected line is y = (1/2)x.But wait, let's check if this line is correct. For instance, take a point on the original line, say (0,3). Its reflection is (2,1), which should lie on the reflected line. (2,1): 1 = (1/2)*2 = 1. Correct. Another point (1,5): reflection is (4,2). 2 = (1/2)*4 = 2. Correct. Also, the intersection point (-2,-1) is on both lines. Checking on the reflected line: y = (1/2)x => -1 = (1/2)*(-2) = -1. Correct.Thus, the equation of the reflected line is y = (1/2)x, which can be written as x - 2y =0. Therefore, the answer should be x - 2y =0. But let me check if the problem requires it in a specific form. The original equation was given as y - 2x -3 =0. So maybe they want it in similar form. Let's write x -2y =0 as -2y +x =0 or 2y -x =0. Alternatively, y = (1/2)x. Any of these forms is correct, but perhaps the standard form is preferred.Alternatively, to check once more, suppose we take the mirror line y =x +1, and the original line y=2x +3. The angle between them is arctan(2) - arctan(1) = arctan(2) - 45 degrees. The reflected line should have an angle on the other side of the mirror line. The slope of the reflected line is 1/2, which is the reciprocal of 2, but not exactly. Wait, reflecting over y =x would swap the coordinates, but here we are reflecting over y =x +1.Wait, but in the translated coordinates, when we shifted down by 1, the mirror line became y =x, and reflection over y=x swaps coordinates, which is why the original line v =2u +2 becomes u=2v +2, which is v = (u -2)/2. Translating back gives the line. So this confirms the result.Another check: the distance from the original line to the mirror line should be equal to the distance from the reflected line to the mirror line. But distance between two parallel lines is constant, but these lines aren't parallel. Wait, but the mirror line is not parallel to either, so maybe the distances vary. Alternatively, since reflection is an isometry, the angles should be preserved. The angle between original line and mirror line should equal the angle between reflected line and mirror line.Original line slope:2, mirror slope:1.Angle between original and mirror:tanθ = |(2 -1)/(1 + 2*1)| = |1/3| => θ = arctan(1/3)Reflected line slope:1/2, mirror slope:1.Angle between reflected and mirror:tanφ = |(1 - 1/2)/(1 +1*(1/2)| = |1/2 / (3/2)| = |1/3| => φ = arctan(1/3)Hence, the angles are equal. Therefore, the reflection is correct.Therefore, I think the answer is correct. Thus, the equation of the symmetric line is x - 2y =0.Final AnswerThe equation of the symmetric line is boxed{x - 2y = 0}.</think>To find the equation of the line symmetric to the line ( y - 2x - 3 = 0 ) with respect to the line ( y = x + 1 ):1. Rewrite the given lines: - The original line is ( y = 2x + 3 ). - The line of symmetry is ( y = x + 1 ).2. Reflect two points on the original line over the line of symmetry: - Choose points on the original line: - Point ( A(0, 3) ) - Point ( B(1, 5) ) - Reflect these points over the line ( y = x + 1 ): - The reflection formula for a point ( (x, y) ) over the line ( x - y + 1 = 0 ) is ( (y - 1, x + 1) ). - Reflecting ( A(0, 3) ): - ( x' = 3 - 1 = 2 ) - ( y' = 0 + 1 = 1 ) - Reflected point: ( (2, 1) ) - Reflecting ( B(1, 5) ): - ( x' = 5 - 1 = 4 ) - ( y' = 1 + 1 = 2 ) - Reflected point: ( (4, 2) )3. Find the equation of the line through the reflected points: - The reflected points are ( (2, 1) ) and ( (4, 2) ). - Calculate the slope ( m ): [ m = frac{2 - 1}{4 - 2} = frac{1}{2} ] - Use the point-slope form with point ( (2, 1) ): [ y - 1 = frac{1}{2}(x - 2) ] - Simplify to get the equation: [ 2(y - 1) = x - 2 implies 2y - 2 = x - 2 implies x - 2y = 0 ]4. Verification: - The reflected line passes through the intersection point of the original line and the mirror line, which is ( (-2, -1) ). - The angles between the original line and the mirror line, and between the reflected line and the mirror line are equal.Thus, the equation of the symmetric line is (boxed{x - 2y = 0}).]

🔑:To solve this, let's take any point A(x, y) on the line we are looking for. The point A symmetric to the line y=x+1 would be B(y-1, x+1), which lies on the line y-2x-3=0. Therefore, we have (x+1)-2(y-1)-3=0. Simplifying this, we get x-2y=0. Hence, the answer is: x-2y=0. By choosing any point A(x, y) on the line we are looking for, and finding its symmetric point B(y-1, x+1) that lies on the line y-2x-3=0, we derive the equation in terms of x and y, which is what we are looking for. This question primarily tests the method of finding the equation of a line that is symmetric to a given line, and it is considered a basic problem.boxed{x-2y=0}

🧠:[<think>Okay, so I need to express (a + b)² in terms of s and t, where s is a - b and t is a³ - b³. Let me start by recalling some algebraic identities that might help here. First, I know that the difference of cubes can be factored: a³ - b³ = (a - b)(a² + ab + b²). Since t is a³ - b³, that means t = s*(a² + ab + b²). So maybe I can relate this to (a + b)² somehow.Wait, (a + b)² expands to a² + 2ab + b². Hmm, the expression for t has a² + ab + b². If I can express a² + ab + b² in terms of (a + b)², that might work. Let me see:a² + ab + b² = (a² + 2ab + b²) - ab = (a + b)² - ab.So substituting back into the expression for t:t = s*( (a + b)² - ab )But I need to express (a + b)² in terms of s and t. Let me denote x = (a + b)². Then the equation becomes t = s*(x - ab). So, t = s*x - s*ab. Hmm, but now there's another term with ab. I need to find a way to express ab in terms of s and t or eliminate it. Let me think. If I can find another equation involving ab, maybe I can solve for ab.I know that s = a - b. Let me square both sides to get s² = (a - b)² = a² - 2ab + b². Also, x = (a + b)² = a² + 2ab + b². So if I add these two equations:s² + x = (a² - 2ab + b²) + (a² + 2ab + b²) = 2a² + 2b² = 2(a² + b²)So, s² + x = 2(a² + b²). Let's call this equation (1).Alternatively, if I subtract them:x - s² = (a² + 2ab + b²) - (a² - 2ab + b²) = 4ab.So, x - s² = 4ab => ab = (x - s²)/4. Let's call this equation (2).Now, going back to the expression for t:t = s*(x - ab) = s*x - s*ab.Substituting ab from equation (2):t = s*x - s*( (x - s²)/4 )Let me simplify this:t = s*x - (s/4)*(x - s²) = s*x - (s*x)/4 + (s³)/4Combine like terms:t = (s*x - s*x/4) + s³/4 = (3s*x)/4 + s³/4Multiply both sides by 4 to eliminate denominators:4t = 3s*x + s³Now, solve for x:3s*x = 4t - s³Divide both sides by 3s:x = (4t - s³)/(3s)Since x is (a + b)², that's the expression we need. Let me check this to make sure.So, (a + b)² = (4t - s³)/(3s). Let me verify with some numbers. Let's pick a = 2 and b = 1. Then s = 2 - 1 = 1, t = 8 - 1 = 7. Then (a + b)^2 = (3)^2 = 9. According to the formula, (4*7 - 1³)/(3*1) = (28 - 1)/3 = 27/3 = 9. Correct.Another example: a = 3, b = 1. Then s = 2, t = 27 - 1 = 26. (a + b)^2 = 16. Formula: (4*26 - 8)/(6) = (104 - 8)/6 = 96/6 = 16. Correct again. Seems like this works.So, the answer should be (4t - s³)/(3s) boxed as per instructions.Final Answerboxed{dfrac{4t - s^3}{3s}}</think>Given distinct real numbers (a) and (b), we need to express ((a + b)^2) in terms of (s = a - b) and (t = a^3 - b^3).First, recall the identity for the difference of cubes:[t = a^3 - b^3 = (a - b)(a^2 + ab + b^2)]Substituting (s = a - b), we get:[t = s(a^2 + ab + b^2)]We know that ((a + b)^2 = a^2 + 2ab + b^2). Let (x = (a + b)^2). Then:[a^2 + ab + b^2 = x - ab]Substituting this into the expression for (t):[t = s(x - ab)]Next, we use the square of (s):[s^2 = (a - b)^2 = a^2 - 2ab + b^2]Adding (s^2) and (x):[s^2 + x = 2(a^2 + b^2)]Subtracting (s^2) from (x):[x - s^2 = 4ab implies ab = frac{x - s^2}{4}]Substituting (ab) back into the expression for (t):[t = sleft(x - frac{x - s^2}{4}right)]Simplifying:[t = sleft(frac{4x - (x - s^2)}{4}right) = sleft(frac{3x + s^2}{4}right)]Multiplying both sides by 4:[4t = 3sx + s^3]Solving for (x):[3sx = 4t - s^3 implies x = frac{4t - s^3}{3s}]Thus, the expression for ((a + b)^2) in terms of (s) and (t) is:[boxed{dfrac{4t - s^3}{3s}}],<think>Alright, let's see. The problem is to express (a + b)² in terms of s and t, where s is a - b and t is a³ - b³. Hmm, okay. So I need to find a relationship between a and b using these two given expressions and then combine them somehow to get (a + b)². First, let me recall some algebraic identities. The difference of cubes formula says that a³ - b³ = (a - b)(a² + ab + b²). Since t is given as a³ - b³, maybe I can use that. Let me write that down:t = a³ - b³ = (a - b)(a² + ab + b²)But we know that s = a - b. So substituting that into the equation, we get:t = s(a² + ab + b²)So, t/s = a² + ab + b², provided that s ≠ 0, which it isn't since a and b are distinct real numbers, so a - b ≠ 0. Now, the target is (a + b)². Let's expand that:(a + b)² = a² + 2ab + b²Comparing this to the expression we have from t/s, which is a² + ab + b², there's a difference in the coefficient of the ab term. The target has 2ab, while t/s has 1ab. So maybe if I can relate these two expressions?Let me denote the expression a² + ab + b² as t/s. Then, the target expression (a + b)² is equal to (a² + ab + b²) + ab. So:(a + b)² = (a² + ab + b²) + ab = (t/s) + abSo now, if I can express ab in terms of s and t, then I can substitute that here. Let's see if we can find ab in terms of s and t.Alternatively, maybe there's another approach. Let me think. We have s = a - b. Let's square both sides:s² = (a - b)² = a² - 2ab + b²So s² = a² + b² - 2abBut from the earlier expression, t/s = a² + ab + b². Let's write that down:t/s = a² + ab + b²So now, we have two equations:1. s² = a² + b² - 2ab2. t/s = a² + ab + b²Let me denote these as Equation 1 and Equation 2.If I can solve these two equations for a² + b² and ab, then perhaps I can find (a + b)².Let me note that (a + b)² = a² + 2ab + b² = (a² + b²) + 2ab. So if I can find expressions for (a² + b²) and ab, then adding them appropriately will give me the desired result.From Equation 1:a² + b² = s² + 2abFrom Equation 2:a² + b² = (t/s) - abSo substituting Equation 1 into Equation 2:s² + 2ab = (t/s) - abNow, let's solve for ab.Bring all terms involving ab to one side:s² = (t/s) - ab - 2abWait, that seems confusing. Let me rearrange the equation properly.Starting from:s² + 2ab = t/s - abSubtract t/s from both sides:s² + 2ab - t/s = -abThen, add ab to both sides:s² + 3ab - t/s = 0Then, move the constants to the other side:3ab = t/s - s²Therefore,ab = (t/s - s²)/3Which can be written as:ab = (t - s³)/(3s)Since t/s is (t)/s, so (t/s - s²) = (t - s³)/s. Then dividing by 3 gives (t - s³)/(3s).Okay, so ab is (t - s³)/(3s). Now, going back to (a + b)². As established earlier:(a + b)² = (a² + b²) + 2abWe have from Equation 1 that a² + b² = s² + 2ab. Wait, but maybe it's better to use the expression from Equation 2. Let me check:From Equation 2: a² + b² = (t/s) - ab. Therefore,(a + b)² = (t/s - ab) + 2ab = t/s + abSo, substituting ab into this:(a + b)² = t/s + (t - s³)/(3s)Combine the terms:= (3t)/(3s) + (t - s³)/(3s)= [3t + t - s³]/(3s)= (4t - s³)/(3s)Hmm, so is that the answer? Let me check my steps again to make sure.First, from the difference of cubes:t = (a - b)(a² + ab + b²) = s*(a² + ab + b²) => a² + ab + b² = t/sThen, s = a - b. Squaring both sides:s² = a² - 2ab + b² => a² + b² = s² + 2abFrom a² + ab + b² = t/s, substitute a² + b² as s² + 2ab:s² + 2ab + ab = t/s => s² + 3ab = t/s => 3ab = t/s - s² => ab = (t/s - s²)/3 = (t - s³)/(3s)Then, (a + b)^2 = a² + 2ab + b² = (a² + ab + b²) + ab = (t/s) + abSubstituting ab:= t/s + (t - s³)/(3s) = (3t + t - s³)/(3s) = (4t - s³)/(3s)So yes, that seems correct. Let me verify with an example. Let me pick some numbers for a and b, compute s and t, then check if (a + b)^2 equals (4t - s³)/(3s).Let’s take a = 2, b = 1. Then s = 2 - 1 = 1. t = 8 - 1 = 7. Then (a + b)^2 = (3)^2 = 9.Compute (4t - s³)/(3s) = (4*7 - 1)/(3*1) = (28 - 1)/3 = 27/3 = 9. That works.Another example. Let a = 3, b = 0. Then s = 3, t = 27. (a + b)^2 = 9. Compute (4*27 - 27)/(3*3) = (108 - 27)/9 = 81/9 = 9. Correct.Another example: a = 5, b = 2. Then s = 3. t = 125 - 8 = 117. (a + b)^2 = 49. Compute (4*117 - 27)/(9) = (468 - 27)/9 = 441/9 = 49. Perfect.Let me try a negative example. Let a = 1, b = -1. Then s = 2. t = 1 - (-1) = 2. (a + b)^2 = 0. Compute (4*2 - 8)/(6) = (8 - 8)/6 = 0/6 = 0. Correct.Hmm, seems like all the test cases check out. So the expression (a + b)² is indeed (4t - s³)/(3s). Therefore, the answer should be boxed{dfrac{4t - s^3}{3s}}.Final Answerboxed{dfrac{4t - s^3}{3s}}</think>Given distinct real numbers (a) and (b), we need to express ((a + b)^2) in terms of (s = a - b) and (t = a^3 - b^3).First, we use the identity for the difference of cubes:[t = a^3 - b^3 = (a - b)(a^2 + ab + b^2)]Since (s = a - b), we substitute (s) into the equation:[t = s(a^2 + ab + b^2)]Solving for (a^2 + ab + b^2), we get:[a^2 + ab + b^2 = frac{t}{s}]Next, we square (s):[s^2 = (a - b)^2 = a^2 - 2ab + b^2]This gives us:[a^2 + b^2 = s^2 + 2ab]We substitute (a^2 + b^2) from the above equation into the expression (a^2 + ab + b^2):[s^2 + 2ab + ab = frac{t}{s}]Simplifying, we get:[s^2 + 3ab = frac{t}{s}]Solving for (ab):[3ab = frac{t}{s} - s^2][ab = frac{frac{t}{s} - s^2}{3} = frac{t - s^3}{3s}]Now, we express ((a + b)^2) using the identity:[(a + b)^2 = a^2 + 2ab + b^2]Substituting (a^2 + b^2 = s^2 + 2ab) into the equation:[(a + b)^2 = (s^2 + 2ab) + 2ab = s^2 + 4ab]Using the value of (ab) found earlier:[(a + b)^2 = s^2 + 4 left( frac{t - s^3}{3s} right)]Simplifying:[(a + b)^2 = s^2 + frac{4(t - s^3)}{3s}]Combining the terms:[(a + b)^2 = frac{3s^3 + 4t - 4s^3}{3s} = frac{4t - s^3}{3s}]Thus, the final answer is:[boxed{dfrac{4t - s^3}{3s}}]]

🔑:## First Solution.We can factorize t=a^{3}-b^{3}=(a-b)left(a^{2}+a b+b^{2}right), i.e., t=sleft(a^{2}+a b+b^{2}right).After dividing by s(s neq 0), we geta^{2}+a b+b^{2}=frac{t}{s}Cubing s=a-b gives s^{3}=a^{3}-3 a^{2} b+3 a b^{2}-b^{3}=a^{3}-b^{3}-3 a b(a-b)=t-3 a b s. Thus, s^{3}=t-3 a b s, from which we can easily expressa b=frac{t-s^{3}}{3 s}Finally, we get(a+b)^{2}=left(a^{2}+a b+b^{2}right)+a b=frac{t}{s}+frac{t-s^{3}}{3 s}=frac{4 t-s^{3}}{3 s}