🧠:[<think>Okay, so I need to find the radius of circle C₄, which is tangent to C₁, C₃, and the line ℓ. The problem mentions that circles C₁ and C₂ have radius 360 and are tangent to each other and the line ℓ. Then there's circle C₃ tangent to C₁, C₂, and ℓ, and then C₄ is tangent to C₁, C₃, and ℓ. Hmm, this seems like a problem involving circles packed between other circles and a line, maybe similar to the Descartes Circle Theorem or something like that.First, let me visualize the setup. There's a horizontal line ℓ. Circles C₁ and C₂ are both sitting above this line, each with radius 360. They're tangent to each other, so the distance between their centers should be twice the radius, which is 720. Since they are both tangent to the line ℓ, their centers are 360 units above ℓ. So if I imagine the line ℓ as the x-axis, then the centers of C₁ and C₂ would be at some points (x, 360). Since they are tangent to each other, the distance between their centers is 720. Let's say C₁ is at (a, 360) and C₂ is at (a + 720, 360). But since they are both tangent to the line ℓ, and the problem doesn't specify their positions left or right, maybe they are placed symmetrically with respect to the y-axis? Wait, actually, since the problem doesn't give specific coordinates, maybe it's better to assign coordinates for simplicity.Let me set the line ℓ as the x-axis. Let's place circle C₁ with its center at (0, 360). Then circle C₂, which is tangent to C₁, should have its center at (720, 360), because the distance between centers is 2*360 = 720. So C₁ is at (0, 360), C₂ at (720, 360). Then circle C₃ is tangent to both C₁, C₂, and ℓ. So C₃ is a circle that sits below C₁ and C₂, touching both of them and the line ℓ. Then circle C₄ is tangent to C₁, C₃, and ℓ. So C₄ would be another circle that's nestled between C₁, C₃, and the line ℓ.To solve this, I think I need to use the method of coordinates or perhaps use the Descartes Circle Theorem, which relates the curvatures (reciprocals of radii) of four mutually tangent circles. However, since we're dealing with a line, which can be considered a circle with infinite radius, curvature 0. Wait, the Descartes Circle Theorem can handle that. Let me recall the formula.The Descartes Circle Theorem states that if four circles are mutually tangent, their curvatures k₁, k₂, k₃, k₄ satisfy:k₄ = k₁ + k₂ + k₃ ± 2√(k₁k₂ + k₂k₃ + k₃k₁)But in our case, we have a line involved. A line can be considered as a circle with curvature 0 (since its radius is infinite). So if three circles are tangent to each other and a line, then we can use a modified version of the Descartes Theorem where one of the curvatures is 0.Let me denote the curvature of a circle as k = 1/r, where r is the radius. For the line ℓ, its curvature is 0. So when dealing with C₃, which is tangent to C₁, C₂, and ℓ, the curvatures would be k₁ = 1/360, k₂ = 1/360, k₃ = 0 (the line), and k₄ = 1/r₃, where r₃ is the radius of C₃.Applying the Descartes Theorem:k₄ = k₁ + k₂ + k₃ ± 2√(k₁k₂ + k₂k₃ + k₃k₁)Plugging in the values:k₄ = (1/360) + (1/360) + 0 ± 2√[(1/360)(1/360) + (1/360)(0) + (0)(1/360)]Simplify:k₄ = 2/360 ± 2√(1/(360)^2)k₄ = 1/180 ± 2*(1/360)k₄ = 1/180 ± 1/180So two possible solutions:k₄ = 1/180 + 1/180 = 2/180 = 1/90, which would mean r₃ = 90Or k₄ = 1/180 - 1/180 = 0, which is the line itself. Since we want the circle tangent to the two circles and the line, we take the positive curvature solution, so k₄ = 1/90, thus r₃ = 90.Wait, but wait, let me check. If the two original circles have radius 360, is the circle between them and the line really going to have radius 90? That seems a bit small, but maybe.Alternatively, maybe there's a formula for circles tangent to two equal circles and a line. Let me think. If two circles of radius R are placed with their centers 2R apart (distance between centers is 2R), then the circle tangent to both and the line below them would have radius r. Let me derive that.The centers of the two big circles are at (0, R) and (2R, R). The small circle is tangent to both and the line y=0. Let its center be at (x, r). Then the distance from (x, r) to (0, R) must be R + r, and the distance from (x, r) to (2R, R) must be R + r. Also, since it's tangent to the line, its y-coordinate is r.So, distance from (x, r) to (0, R):√[(x - 0)^2 + (r - R)^2] = R + rSquaring both sides:x² + (r - R)² = (R + r)²Expand:x² + r² - 2Rr + R² = R² + 2Rr + r²Simplify:x² - 2Rr = 2Rrx² = 4RrSimilarly, distance from (x, r) to (2R, R):√[(x - 2R)^2 + (r - R)^2] = R + rSquaring:(x - 2R)^2 + (r - R)^2 = (R + r)^2Expand left side:x² - 4R x + 4R² + r² - 2Rr + R²Right side:R² + 2Rr + r²So:x² -4R x + 4R² + r² - 2Rr + R² = R² + 2Rr + r²Simplify:x² -4R x + 4R² + (-2Rr) = 2RrBring terms over:x² -4R x + 4R² -4Rr = 0But from earlier, x² = 4Rr. Substitute x² into this equation:4Rr -4R x + 4R² -4Rr = 0Simplify:-4R x + 4R² = 0Divide by -4R:x - R = 0 => x = RSo x = R. Then from x² = 4Rr, we have R² = 4Rr => R = 4r => r = R/4Since R is 360, r = 360/4 = 90. So that's consistent with the Descartes Theorem result. So C₃ has radius 90. Okay, so that seems correct.Now, moving on to C₄, which is tangent to C₁, C₃, and ℓ. So we need to find the radius of C₄. Let's denote its radius as r₄.Again, we can use the Descartes Circle Theorem here, considering that C₄ is tangent to C₁ (radius 360), C₃ (radius 90), and the line ℓ (curvature 0). Let's apply the theorem again.Let k₁ = 1/360, k₂ = 1/90, k₃ = 0 (the line), and k₄ = 1/r₄.Applying the formula:k₄ = k₁ + k₂ + k₃ ± 2√(k₁k₂ + k₂k₃ + k₃k₁)Plugging in:k₄ = (1/360) + (1/90) + 0 ± 2√[(1/360)(1/90) + (1/90)(0) + (0)(1/360)]Simplify:First, calculate the sum:1/360 + 1/90 = 1/360 + 4/360 = 5/360 = 1/72So k₄ = 1/72 ± 2√[(1/(360*90)) + 0 + 0]Calculate the square root term:√[1/(360*90)] = √[1/(32400)] = 1/180Multiply by 2:2*(1/180) = 1/90Therefore, k₄ = 1/72 ± 1/90Compute both possibilities:1) k₄ = 1/72 + 1/90To add these, find a common denominator. 72 and 90 have LCM of 360.1/72 = 5/3601/90 = 4/360So 5/360 + 4/360 = 9/360 = 1/402) k₄ = 1/72 - 1/90Similarly:5/360 - 4/360 = 1/360Therefore, k₄ could be 1/40 or 1/360.But we need to determine which curvature is the correct one. Since we already have C₃ with curvature 1/90, and C₁ with 1/360, and the line, the new circle C₄ should be smaller than C₃, right? Because it's nestled between C₁, C₃, and the line. If C₃ has radius 90, then C₄ should be smaller. So let's check the two solutions.If k₄ = 1/40, then r₄ = 40. If k₄ = 1/360, then r₄ = 360. But 360 is the same as C₁, which doesn't make sense because C₄ is supposed to be tangent to C₁, C₃, and the line. So the radius 360 would mean it's the same as C₁, which can't be. Therefore, the correct curvature is 1/40, so r₄ = 40. Wait, but 40 is smaller than 90, which makes sense. So the answer is 40? Hmm.But let me verify this with coordinate geometry as well, to be sure.Let's model the problem again. The centers of C₁, C₃, and ℓ. Let me assign coordinates. Let’s set the line ℓ as the x-axis. Then the center of C₁ is at (0, 360). The center of C₃ we previously found at (R, r₃) where R = 360, and r₃ = 90, so center at (360, 90). Wait, hold on, in the previous problem where we had two circles of radius 360, their centers were at (0, 360) and (720, 360). Then the center of C₃ was at (360, 90). Because x was R, which was 360. Wait, in the previous calculation, when we set R = 360, then x = R = 360, and r = 90. So the center of C₃ is at (360, 90). So C₃ is centered at (360, 90) with radius 90. Then circle C₄ is tangent to C₁ (center (0, 360), radius 360), C₃ (center (360, 90), radius 90), and the line ℓ (y=0). So let's find the radius of C₄, let's call it r₄. The center of C₄ will be at (h, r₄), since it's tangent to the x-axis. Then, the distance from (h, r₄) to (0, 360) should be 360 + r₄, and the distance from (h, r₄) to (360, 90) should be 90 + r₄.So, set up the equations:√[(h - 0)^2 + (r₄ - 360)^2] = 360 + r₄ ...(1)√[(h - 360)^2 + (r₄ - 90)^2] = 90 + r₄ ...(2)Let me square both equations to remove the square roots.Equation (1):h² + (r₄ - 360)^2 = (360 + r₄)^2Expand both sides:h² + r₄² - 720r₄ + 360² = r₄² + 720r₄ + 360²Simplify:h² - 720r₄ = 720r₄h² = 1440r₄ ...(3)Equation (2):(h - 360)^2 + (r₄ - 90)^2 = (90 + r₄)^2Expand left side:(h² - 720h + 360²) + (r₄² - 180r₄ + 90²)Right side:r₄² + 180r₄ + 90²So:h² -720h + 360² + r₄² -180r₄ + 90² = r₄² + 180r₄ + 90²Cancel terms:h² -720h + 360² -180r₄ = 180r₄Bring terms to one side:h² -720h + 360² -180r₄ -180r₄ = 0Simplify:h² -720h + 360² -360r₄ = 0Now, substitute h² from equation (3):1440r₄ -720h + 360² -360r₄ = 0Simplify:(1440r₄ -360r₄) + (-720h) + 360² = 01080r₄ -720h + 360² = 0Divide through by 360 to simplify:3r₄ - 2h + 360 = 0So:3r₄ = 2h - 360 ...(4)From equation (3), h² = 1440r₄. Let's solve for h in terms of r₄ from equation (4):From equation (4):2h = 3r₄ + 360 => h = (3r₄ + 360)/2Substitute h into equation (3):[(3r₄ + 360)/2]^2 = 1440r₄Expand left side:[9r₄² + 2*3r₄*360 + 360²]/4 = 1440r₄Multiply both sides by 4:9r₄² + 2160r₄ + 129600 = 5760r₄Bring all terms to left side:9r₄² + 2160r₄ + 129600 -5760r₄ = 0Simplify:9r₄² - 3600r₄ + 129600 = 0Divide through by 9:r₄² - 400r₄ + 14400 = 0Quadratic equation: r₄² -400r₄ +14400 = 0Solve using quadratic formula:r₄ = [400 ± √(400² -4*1*14400)] / 2Calculate discriminant:400² -4*1*14400 = 160000 -57600 = 102400√102400 = 320Thus:r₄ = [400 ±320]/2Compute both solutions:r₄ = (400 +320)/2 = 720/2 = 360r₄ = (400 -320)/2 =80/2 =40So r₄ = 360 or 40. But r₄=360 would be the same as C₁, which is not possible since C₄ is tangent to C₁ and the line. So the valid solution is r₄=40. So the radius of C₄ is 40. That matches the Descartes Theorem result earlier. Therefore, the answer should be 40. But let me confirm once again.Wait, if C₄ has radius 40, then its center is at (h, 40). From equation (3), h²=1440*40=57600. So h=√57600=240. Therefore, center at (240,40). Let's check distances.Distance from (240,40) to C₁'s center (0,360):√[(240-0)^2 + (40-360)^2] = √[57600 + (-320)^2] = √[57600 +102400] = √160000=400. Since 360 +40=400, which matches.Distance from (240,40) to C₃'s center (360,90):√[(240-360)^2 + (40-90)^2] = √[(-120)^2 + (-50)^2] = √[14400 +2500]=√16900=130. Since 90 +40=130, which also matches. Perfect, that works. So the radius of C₄ is indeed 40.Therefore, the answer is 40. So I think that's correct.Final AnswerThe radius of ( C_4 ) is (boxed{40}).</think>To find the radius of circle ( C_4 ), we start by noting that circles ( C_1 ) and ( C_2 ) each have a radius of 360 and are tangent to each other and the line ( ell ). Circle ( C_3 ) is tangent to ( C_1 ), ( C_2 ), and ( ell ), and circle ( C_4 ) is tangent to ( C_1 ), ( C_3 ), and ( ell ).First, we use the Descartes Circle Theorem to find the radius of ( C_3 ). The theorem states that for circles tangent to each other and a line (curvature 0), the curvature ( k ) (reciprocal of radius) of the new circle can be found using:[k_4 = k_1 + k_2 + k_3 pm 2sqrt{k_1k_2 + k_2k_3 + k_3k_1}]For ( C_3 ), the curvatures are ( k_1 = frac{1}{360} ), ( k_2 = frac{1}{360} ), and ( k_3 = 0 ). Plugging these into the theorem:[k_4 = frac{1}{360} + frac{1}{360} + 0 pm 2sqrt{left(frac{1}{360} cdot frac{1}{360}right) + 0 + 0}][k_4 = frac{1}{180} pm 2 cdot frac{1}{360}][k_4 = frac{1}{180} pm frac{1}{180}]Choosing the positive solution (since the curvature must be positive for a circle tangent to the line and the two circles), we get ( k_4 = frac{1}{90} ), so the radius ( r_3 = 90 ).Next, we apply the Descartes Circle Theorem again to find the radius of ( C_4 ). The curvatures are ( k_1 = frac{1}{360} ), ( k_3 = frac{1}{90} ), and ( k_3 = 0 ):[k_4 = frac{1}{360} + frac{1}{90} + 0 pm 2sqrt{left(frac{1}{360} cdot frac{1}{90}right) + 0 + 0}][k_4 = frac{1}{72} pm 2 cdot frac{1}{180}][k_4 = frac{1}{72} pm frac{1}{90}]Converting to a common denominator:[k_4 = frac{5}{360} + frac{4}{360} = frac{9}{360} = frac{1}{40} quad text{(positive solution)}]Thus, the radius ( r_4 = 40 ).Verification using coordinate geometry confirms the radius of ( C_4 ) is indeed 40. The center of ( C_4 ) is calculated to be at (240, 40), with distances to ( C_1 ) and ( C_3 ) matching the required tangency conditions.The radius of ( C_4 ) is (boxed{40}).,<think>Okay, let's see. So there's this geometry problem with circles tangent to each other and a line. I need to find the radius of circle C4. Hmm. Let me start by visualizing the problem. There's a line ℓ, and two circles C1 and C2, each with radius 360, tangent to each other and both tangent to the line. Then there's another circle C3 that's tangent to C1, C2, and ℓ. Then C4 is tangent to C1, C3, and ℓ. So, I need to find the radius of C4.First, maybe I should draw a diagram. But since I can't actually draw, I'll imagine it. The line ℓ is horizontal, let's say. C1 and C2 are both sitting on ℓ, touching it. Since their radius is 360, their centers are 360 units above ℓ. Also, they are tangent to each other, so the distance between their centers is twice the radius, which is 720. So if I consider the centers of C1 and C2, they are 720 units apart horizontally and 360 units above ℓ. That makes sense.Now, circle C3 is tangent to both C1, C2, and ℓ. So C3 is sitting below the line connecting C1 and C2, touching both of them and the line ℓ. This is probably a smaller circle. Similarly, C4 is tangent to C1, C3, and ℓ, so it's even smaller, nestled between C1, C3, and ℓ.This seems like a problem that can be approached using the method of coordinates or maybe using Descartes' Circle Theorem. Since there are multiple tangent circles, Descartes' Theorem might be useful here. Let me recall what that is.Descartes' Circle Theorem states that if four circles are mutually tangent, their curvatures satisfy a certain equation. But in our case, we have three circles and a line. However, a line can be considered a circle with infinite radius, so its curvature is zero. The curvature is defined as 1/radius. So, if we have three circles and a line, we can apply Descartes' Theorem by treating the line as a circle with curvature zero.Let me write down the theorem. If four circles are mutually tangent, then:k4 = k1 + k2 + k3 ± 2√(k1k2 + k2k3 + k3k1)where k is the curvature (1/r). But when one of the circles is a line, its curvature k is 0. So, let's apply this for the case of C1, C2, ℓ, and C3.First, for C3. The curvatures of C1 and C2 are both 1/360. The curvature of ℓ is 0. Let me denote the curvature of C3 as k3. Then according to Descartes' Theorem:k3 = k1 + k2 + k_line ± 2√(k1k2 + k2k_line + k_linek1)But since k_line is 0, this simplifies to:k3 = (1/360 + 1/360 + 0) ± 2√[(1/360)(1/360) + (1/360)(0) + (0)(1/360)]Simplify the terms inside:k3 = (2/360) ± 2√[1/(360^2)]Which is:k3 = (1/180) ± 2*(1/360)Wait, sqrt(1/360^2) is 1/360, so 2*(1/360) is 1/180.Therefore:k3 = (1/180) ± (1/180)So, two possibilities:k3 = 1/180 + 1/180 = 1/90ork3 = 1/180 - 1/180 = 0But curvature 0 is the line itself, which we already have. So, the new curvature must be 1/90. Therefore, the radius of C3 is 90. Wait, that seems too small? Let me check.Wait, Descartes' Theorem gives two solutions, one is the line (which we discard) and the other is the circle. But in this case, the line is already part of the configuration. So, maybe the curvature calculation is correct? If the radius is 90, that would mean C3 has radius 90. Let me verify with coordinates.Alternatively, maybe I made a mistake in the application. Let's try coordinate geometry.Let me set up a coordinate system. Let the line ℓ be the x-axis. Then, the centers of C1 and C2 are at (x1, 360) and (x2, 360). Since the circles are tangent to each other, the distance between centers is 720. Let's set C1 at (0, 360) and C2 at (720, 360). Then, circle C3 is tangent to both C1, C2, and the x-axis. Let the center of C3 be (h, k), with radius r. Since it's tangent to the x-axis, the y-coordinate k is equal to r. So the center is (h, r).Now, the distance between C3 and C1 must be equal to 360 + r. So:sqrt((h - 0)^2 + (r - 360)^2) = 360 + rSimilarly, the distance between C3 and C2 is sqrt((h - 720)^2 + (r - 360)^2) = 360 + rSo, these two equations:h^2 + (r - 360)^2 = (360 + r)^2and(h - 720)^2 + (r - 360)^2 = (360 + r)^2Let me expand the first equation:h^2 + (r^2 - 720r + 360^2) = r^2 + 720r + 360^2Simplify:h^2 - 720r = 720rTherefore:h^2 = 1440rSimilarly, the second equation:(h - 720)^2 + (r - 360)^2 = (360 + r)^2Expand:h^2 - 1440h + 720^2 + r^2 - 720r + 360^2 = r^2 + 720r + 360^2Cancel out terms:h^2 - 1440h + 720^2 - 720r = 720rBring all terms to left:h^2 - 1440h + 720^2 - 720r - 720r = 0Simplify:h^2 - 1440h + 720^2 - 1440r = 0But from the first equation, h^2 = 1440r, so substitute h^2:1440r - 1440h + 720^2 - 1440r = 0Simplify:-1440h + 720^2 = 0Therefore:-1440h + 518400 = 0So, 1440h = 518400Divide both sides by 1440:h = 518400 / 1440 = 360Wait, h = 360. Then from h^2 = 1440r, so 360^2 = 1440rSo, r = (360^2)/1440 = (129600)/1440 = 90. So, yes, r is 90. So C3 has radius 90.Okay, so Descartes' Theorem gave me the same result. So that's good. So, radius of C3 is 90. Now, moving on to C4, which is tangent to C1, C3, and ℓ. So, similar problem, but now instead of C2, it's C3. Let's apply Descartes' Theorem again.So, this time, the three "circles" are C1 (radius 360), C3 (radius 90), and ℓ (curvature 0). Let's denote the curvature of C4 as k4. Then, according to Descartes' Theorem:k4 = k1 + k3 + k_line ± 2√(k1k3 + k3k_line + k_linek1)Again, k_line = 0, so:k4 = (1/360 + 1/90 + 0) ± 2√[(1/360)(1/90) + 0 + 0]Compute the terms:First, 1/360 + 1/90 = 1/360 + 4/360 = 5/360 = 1/72Second, inside the square root: (1/360)(1/90) = 1/(360*90) = 1/32400So, sqrt(1/32400) = 1/180, so 2*(1/180) = 1/90Therefore, k4 = 1/72 ± 1/90Compute both possibilities:1/72 + 1/90 = (5 + 4)/360 = 9/360 = 1/401/72 - 1/90 = (5 - 4)/360 = 1/360So, curvature k4 is either 1/40 or 1/360. 1/360 would correspond to a circle with radius 360, which is C1 or C2, so the new circle must be the other solution. Hence, k4 = 1/40, so radius is 40. Wait, but let me verify with coordinate geometry again.Alternatively, maybe coordinate geometry is more reliable here.Let me set up coordinates again. C1 is at (0, 360), C3 is at (360, 90) since earlier we found h = 360 and r = 90. Wait, hold on. Wait, when we calculated C3, its center was at (h, r) = (360, 90). So, coordinates (360, 90). Now, we need to find circle C4, tangent to C1, C3, and ℓ.So, similar to the previous problem. Let the center of C4 be (h4, r4), since it's tangent to ℓ, so y-coordinate is r4. Then, the distance from C4 to C1 must be 360 + r4, and the distance from C4 to C3 must be 90 + r4.So:sqrt((h4 - 0)^2 + (r4 - 360)^2) = 360 + r4andsqrt((h4 - 360)^2 + (r4 - 90)^2) = 90 + r4Let me square both equations.First equation:h4^2 + (r4 - 360)^2 = (360 + r4)^2Expand:h4^2 + r4^2 - 720r4 + 360^2 = r4^2 + 720r4 + 360^2Cancel terms:h4^2 - 720r4 = 720r4So:h4^2 = 1440r4Second equation:(h4 - 360)^2 + (r4 - 90)^2 = (90 + r4)^2Expand:h4^2 - 720h4 + 360^2 + r4^2 - 180r4 + 90^2 = r4^2 + 180r4 + 90^2Cancel terms:h4^2 - 720h4 + 360^2 - 180r4 = 180r4Bring all terms to left:h4^2 - 720h4 + 360^2 - 180r4 - 180r4 = 0Simplify:h4^2 - 720h4 + 360^2 - 360r4 = 0Again, from the first equation, h4^2 = 1440r4. Substitute:1440r4 - 720h4 + 360^2 - 360r4 = 0Simplify:(1440r4 - 360r4) -720h4 + 360^2 = 01080r4 -720h4 + 129600 = 0Divide all terms by 360 to simplify:3r4 - 2h4 + 360 = 0So, 3r4 = 2h4 - 360But from h4^2 = 1440r4, we can express h4 in terms of r4 or vice versa. Let's solve for h4 from h4^2 = 1440r4 => h4 = sqrt(1440r4). However, since the problem is symmetric, perhaps h4 is aligned with the previous centers? Wait, C3 was at (360,90), which is halfway between C1 and C2. But C4 is tangent to C1 and C3, so maybe it's not in the middle. Hmm.Alternatively, let's use substitution. From 3r4 = 2h4 - 360, so h4 = (3r4 + 360)/2Plug this into h4^2 = 1440r4:[(3r4 + 360)/2]^2 = 1440r4Expand left side:(9r4^2 + 2160r4 + 129600)/4 = 1440r4Multiply both sides by 4:9r4^2 + 2160r4 + 129600 = 5760r4Bring all terms to left:9r4^2 + 2160r4 + 129600 - 5760r4 = 0Combine like terms:9r4^2 - 3600r4 + 129600 = 0Divide by 9:r4^2 - 400r4 + 14400 = 0Quadratic equation: r4^2 -400r4 +14400=0Use quadratic formula:r4 = [400 ± sqrt(400^2 - 4*1*14400)]/2Calculate discriminant:160000 - 57600 = 102400sqrt(102400) = 320Therefore,r4 = [400 ± 320]/2So,r4 = (400 + 320)/2 = 720/2 = 360orr4 = (400 - 320)/2 = 80/2 = 40But r4 can't be 360, since that's the radius of C1. So, r4 = 40. Therefore, the radius of C4 is 40. This matches the Descartes' Theorem result. So, answer is 40.Wait, but let me check if this makes sense. If C3 is 90, then C4 is 40, which is smaller. So, the radius is decreasing by a factor. From 360 to 90, which is 1/4, then to 40, which is roughly half of 90? Wait, not exactly a factor. But in Descartes' Theorem, when you have two circles and a line, each subsequent circle's radius can be found by that formula. Let me check with Descartes' again.Wait, when we applied Descartes' Theorem the first time, we had k3 = 1/90. Then, for C4, with k1 = 1/360, k3 = 1/90, and k_line = 0, we got k4 = 1/40, so radius 40, which is consistent. So, that seems okay.Alternatively, maybe there's a pattern here. The radii are 360, 90, 40. Let me see if these follow a harmonic progression or something. But 360 to 90 is dividing by 4, 90 to 40 is dividing by 2.25. Hmm, not sure. Maybe each time the radius is a quarter of the previous? Not exactly. But in any case, the calculation seems consistent both via Descartes' Theorem and coordinate geometry. So, 40 is the answer.So, I think the radius of C4 is 40.Final Answerboxed{40}</think>To solve the problem, we start by noting that circles ( C_1 ) and ( C_2 ) each have a radius of 360 and are tangent to each other and the line ( ell ). We need to find the radius of circle ( C_4 ), which is tangent to ( C_1 ), ( C_3 ), and ( ell ).First, we use Descartes' Circle Theorem to find the radius of ( C_3 ). Descartes' Theorem states that for four mutually tangent circles with curvatures ( k_1, k_2, k_3, k_4 ), the following holds:[k_4 = k_1 + k_2 + k_3 pm 2sqrt{k_1k_2 + k_2k_3 + k_3k_1}]Treating the line ( ell ) as a circle with curvature 0, the curvatures for ( C_1 ) and ( C_2 ) are ( frac{1}{360} ). Applying Descartes' Theorem:[k_3 = frac{1}{360} + frac{1}{360} + 0 pm 2sqrt{left(frac{1}{360} cdot frac{1}{360}right) + 0 + 0}]Simplifying, we get:[k_3 = frac{2}{360} pm 2 cdot frac{1}{360} = frac{1}{180} pm frac{1}{180}]The valid solution for ( k_3 ) (excluding the line) is ( frac{1}{90} ), giving a radius of 90 for ( C_3 ).Next, we apply Descartes' Theorem again to find the radius of ( C_4 ), which is tangent to ( C_1 ), ( C_3 ), and ( ell ). The curvatures for ( C_1 ) and ( C_3 ) are ( frac{1}{360} ) and ( frac{1}{90} ), respectively. Applying the theorem:[k_4 = frac{1}{360} + frac{1}{90} + 0 pm 2sqrt{left(frac{1}{360} cdot frac{1}{90}right) + 0 + 0}]Simplifying, we get:[k_4 = frac{1}{72} pm 2 cdot frac{1}{180} = frac{1}{72} pm frac{1}{90}]The valid solution for ( k_4 ) (excluding the existing circles) is ( frac{1}{40} ), giving a radius of 40 for ( C_4 ).Thus, the radius of ( C_4 ) is (boxed{40}).]

🔑:25. Answer: 40 .Let R be the radius of C_{3}. Then(360-R)^{2}+360^{2}=(360+R)^{2} Rightarrow R=90 .Let r be the radius of C_{4}. Thensqrt{(360+r)^{2}-(360-r)^{2}}+sqrt{(90+r)^{2}-(90-r)^{2}}=360 Rightarrow r=40

🧠:[<think>Okay, let's see. I have this problem here: If (2^{m} = 5) and (5^{n} = 2), then I need to find the value of (frac{1}{m+1} + frac{1}{n+1}). The answer choices are A: 1/2, B: 1, C: 1/3, D: 2. Hmm. Alright, let's start by understanding what's given here.First, (2^{m} = 5). So, m is the exponent to which we raise 2 to get 5. Similarly, (5^{n} = 2), meaning n is the exponent to which we raise 5 to get 2. So, m and n are logarithms here. That makes sense because logarithms are exponents. Specifically, m is (log_{2}5) and n is (log_{5}2). Wait, so maybe there's a relationship between m and n? Since they're logarithms with bases swapped.I remember that (log_{a}b = frac{1}{log_{b}a}). Is that correct? Let me check. If I have (log_{a}b = x), then (a^{x} = b). Taking logarithm base b of both sides: (log_{b}a^{x} = log_{b}b). Which is (x cdot log_{b}a = 1), so (x = frac{1}{log_{b}a}). Yes, that's right. Therefore, (log_{a}b) is the reciprocal of (log_{b}a). So, in this case, m = (log_{2}5) and n = (log_{5}2), so m and n are reciprocals of each other. Therefore, m * n = 1. Is that correct? Let's verify.Given m = (log_{2}5), so n = (log_{5}2). Then, m * n = (log_{2}5 * log_{5}2). Let's compute that. Let me recall the change of base formula: (log_{a}b = frac{ln b}{ln a}). So, (log_{2}5 = frac{ln 5}{ln 2}), and (log_{5}2 = frac{ln 2}{ln 5}). Multiplying these two gives (frac{ln 5}{ln 2} * frac{ln 2}{ln 5} = 1). Yes, that's right. So m * n = 1. That's a useful relationship. So n = 1/m. Or m = 1/n. Therefore, they are reciprocals.So, if m * n = 1, then n = 1/m. Therefore, maybe we can express everything in terms of m or in terms of n. Let's see. The expression we need to compute is (frac{1}{m+1} + frac{1}{n+1}). Since n = 1/m, substituting that into the expression gives:(frac{1}{m + 1} + frac{1}{(1/m) + 1}). Let me compute that. First, let's compute the second term: (frac{1}{(1/m) + 1}). Let's write the denominator as a single fraction: (frac{1 + m}{m}). Therefore, the reciprocal is (frac{m}{1 + m}). So, the second term is (frac{m}{m + 1}). Therefore, the entire expression becomes (frac{1}{m + 1} + frac{m}{m + 1}). Since the denominators are the same, we can add the numerators: (1 + m) over (m + 1), which simplifies to (frac{1 + m}{1 + m} = 1). So, the answer is 1, which is option B. Wait, that seems straightforward. Did I do that correctly?Let me check again. If m * n = 1, then n = 1/m. Substitute into the expression:First term: 1/(m + 1)Second term: 1/(n + 1) = 1/(1/m + 1) = 1/( (1 + m)/m ) = m/(1 + m)Adding them together: [1 + m]/(1 + m) = 1. Yep, that checks out. So, the answer is B: 1. Hmm, that seems too easy. Maybe I made a mistake?Wait, let's try plugging in actual numbers. Let me compute m and n numerically and see if the expression gives 1.Given (2^m = 5), so m = log base 2 of 5. Using a calculator, log2(5) is approximately 2.321928095. Then n = log base 5 of 2, which is approximately 0.4306765581. Let's check if m * n is approximately 1: 2.321928095 * 0.4306765581 ≈ 1.0000000. So that's correct.Now compute 1/(m + 1) + 1/(n + 1). First, m + 1 ≈ 2.321928095 + 1 = 3.321928095. So 1/(m + 1) ≈ 1/3.321928095 ≈ 0.3000. Then n + 1 ≈ 0.4306765581 + 1 = 1.4306765581. 1/(n + 1) ≈ 1/1.4306765581 ≈ 0.7000. Adding them together: 0.3000 + 0.7000 ≈ 1.0000. Wow, exactly 1. So numerically, that also checks out. Therefore, the answer is indeed 1. So, option B.But let me try to think if there's another way to approach this problem, maybe using exponent rules or logarithms.Given that (2^m = 5), take both sides to the power of n: ( (2^m)^n = 5^n ). But (5^n = 2), so (2^{m n} = 2^1). Therefore, m n = 1. So, again, m n = 1.So, the expression is ( frac{1}{m + 1} + frac{1}{n + 1} ). Let me express this as:[frac{1}{m + 1} + frac{1}{n + 1} = frac{n + 1 + m + 1}{(m + 1)(n + 1)} = frac{m + n + 2}{(m + 1)(n + 1)}]But since m n = 1, let's compute the denominator:(m + 1)(n + 1) = m n + m + n + 1 = 1 + m + n + 1 = m + n + 2So numerator is m + n + 2, denominator is m + n + 2. Therefore, the expression is equal to 1. Therefore, the answer is 1. So that's another way to see it. So regardless of the values of m and n, as long as their product is 1, the expression is 1.Therefore, the answer is B: 1.Final Answerboxed{B}</think>Given (2^m = 5) and (5^n = 2), we need to find the value of (frac{1}{m+1} + frac{1}{n+1}).First, we recognize that (m = log_2 5) and (n = log_5 2). By the property of logarithms, (log_a b = frac{1}{log_b a}), we know that (m) and (n) are reciprocals of each other. Therefore, (m cdot n = 1).Substituting (n = frac{1}{m}) into the expression (frac{1}{m+1} + frac{1}{n+1}):[frac{1}{m + 1} + frac{1}{frac{1}{m} + 1}]Simplifying the second term:[frac{1}{frac{1}{m} + 1} = frac{1}{frac{1 + m}{m}} = frac{m}{1 + m}]Thus, the expression becomes:[frac{1}{m + 1} + frac{m}{m + 1} = frac{1 + m}{m + 1} = 1]We can also verify this by considering the numerator and denominator of the combined fraction:[frac{1}{m + 1} + frac{1}{n + 1} = frac{m + n + 2}{(m + 1)(n + 1)}]Since (m cdot n = 1), the denominator expands to:[(m + 1)(n + 1) = mn + m + n + 1 = 1 + m + n + 1 = m + n + 2]Thus, the expression simplifies to:[frac{m + n + 2}{m + n + 2} = 1]Therefore, the value of (frac{1}{m+1} + frac{1}{n+1}) is (boxed{B}).,<think>Okay, let's see. The problem says if 2^m = 5 and 5^n = 2, then find the value of 1/(m+1) + 1/(n+1). The options are A: 1/2, B:1, C:1/3, D:2. Hmm, so we need to find m and n first, maybe? Or is there a way to relate them without finding their exact values? Let's think step by step.First, given that 2^m = 5. So, m is the exponent you need to raise 2 to get 5. Similarly, 5^n = 2, so n is the exponent you need to raise 5 to get 2. That sounds like logarithms. Let me recall that if a^b = c, then b = log_a(c). So, m should be log base 2 of 5, and n should be log base 5 of 2. So, m = log2(5) and n = log5(2). Wait, there's a relationship between logarithms and their reciprocals. Specifically, log_b(a) = 1 / log_a(b). So, log2(5) is equal to 1 / log5(2). Therefore, m = 1/n. Because m is log2(5) and n is log5(2), so m = 1/n. Therefore, m * n = 1. That's an important relation. So, m * n = 1. That might come in handy.Now, the expression we need to evaluate is 1/(m + 1) + 1/(n + 1). Let's write that down:1/(m + 1) + 1/(n + 1)Let me combine the fractions. The common denominator is (m + 1)(n + 1). So, combining them:[ (n + 1) + (m + 1) ] / [ (m + 1)(n + 1) ) ]Simplify numerator:(n + 1 + m + 1) = (m + n + 2)Denominator:(m + 1)(n + 1) = mn + m + n + 1But earlier, we found that mn = 1, since m = 1/n. Therefore, mn = 1. So substitute that in:Denominator: 1 + m + n + 1 = m + n + 2Therefore, the entire expression becomes:(m + n + 2) / (m + n + 2) = 1So, the value is 1. Which is option B. Wait, that seems straightforward. Let me check again to make sure I didn't make a mistake.So, if m * n = 1, then denominator becomes mn + m + n + 1 = 1 + m + n + 1 = m + n + 2, which is same as the numerator. Therefore, the fraction is 1. That's correct.Alternatively, maybe try plugging in actual numbers. Let's see. If 2^m = 5, then m is approximately log2(5). Let me calculate that. log2(5) is about 2.3219. Then n is log5(2), which is 1 / log2(5) ≈ 1 / 2.3219 ≈ 0.4307. Then, m + 1 ≈ 2.3219 + 1 ≈ 3.3219, so 1/(m + 1) ≈ 1/3.3219 ≈ 0.300. Then n + 1 ≈ 0.4307 + 1 ≈ 1.4307, so 1/(n + 1) ≈ 1/1.4307 ≈ 0.699. Adding those together: 0.300 + 0.699 ≈ 0.999, which is approximately 1. So, that checks out numerically. Therefore, the answer should be 1, which is option B.But just to be thorough, let's try another approach. Let's express m and n in terms of logarithms with the same base. Let's pick natural logarithm.Given 2^m = 5. Taking natural log on both sides: m * ln(2) = ln(5), so m = ln(5)/ln(2). Similarly, 5^n = 2. Taking natural log: n * ln(5) = ln(2), so n = ln(2)/ln(5). Therefore, m = ln(5)/ln(2), n = ln(2)/ln(5). Then, m * n = [ln(5)/ln(2)] * [ln(2)/ln(5)] = 1, which confirms our earlier result.Now, let's compute 1/(m + 1) + 1/(n + 1). Let's denote m = ln(5)/ln(2). Then n = ln(2)/ln(5). Let's compute m + 1 and n + 1:m + 1 = [ln(5)/ln(2)] + 1 = [ln(5) + ln(2)] / ln(2) = ln(5*2)/ln(2) = ln(10)/ln(2)Similarly, n + 1 = [ln(2)/ln(5)] + 1 = [ln(2) + ln(5)] / ln(5) = ln(10)/ln(5)Therefore, 1/(m + 1) = ln(2)/ln(10) and 1/(n + 1) = ln(5)/ln(10)Adding them together: [ln(2) + ln(5)] / ln(10) = ln(10)/ln(10) = 1. That's another way to see it. So, yes, the answer is 1. So, option B is correct. This seems to hold true both algebraically and numerically. Therefore, I can be confident the answer is B.Final Answerboxed{B}</think>Given (2^m = 5) and (5^n = 2), we need to find the value of (frac{1}{m+1} + frac{1}{n+1}).First, we recognize that (m = log_2 5) and (n = log_5 2). Using the property of logarithms, we know that (log_b a = frac{1}{log_a b}), so (m = frac{1}{n}) and (n = frac{1}{m}). Therefore, (m cdot n = 1).Next, we consider the expression (frac{1}{m+1} + frac{1}{n+1}). Combining the fractions over a common denominator, we get:[frac{1}{m+1} + frac{1}{n+1} = frac{(n+1) + (m+1)}{(m+1)(n+1)}]Simplifying the numerator and denominator:[frac{m + n + 2}{(m+1)(n+1)}]Expanding the denominator:[(m+1)(n+1) = mn + m + n + 1]Since (mn = 1), the denominator becomes:[1 + m + n + 1 = m + n + 2]Thus, the expression simplifies to:[frac{m + n + 2}{m + n + 2} = 1]Therefore, the value of (frac{1}{m+1} + frac{1}{n+1}) is (boxed{B}).]

🔑:Given that 2^{m}=5 and 5^{n}=2, we aim to find the value of frac{1}{m+1}+frac{1}{n+1}.First, we express m and n in terms of logarithms:- m=log_{2}5- n=log_{5}2Next, we substitute these expressions into the given formula:[frac{1}{m+1}+frac{1}{n+1} = frac{1}{log_{2}5+1}+frac{1}{log_{5}2+1}]We can simplify each term by incorporating the property of logarithms that log_{a}b + log_{a}a = log_{a}(b cdot a):[= frac{1}{log_{2}5+log_{2}2}+frac{1}{log_{5}2+log_{5}5}]This simplifies further because log_{2}2 = 1 and log_{5}5 = 1, leading to:[= frac{1}{log_{2}(5 cdot 2)}+frac{1}{log_{5}(2 cdot 5)}]Since 5 cdot 2 = 10, we have:[= frac{1}{log_{2}10}+frac{1}{log_{5}10}]Using the change of base formula, log_{a}b = frac{1}{log_{b}a}, we can rewrite the expression as:[= log_{10}2+log_{10}5]Combining the logarithms, we get:[= log_{10}(2 times 5)]Simplifying the expression inside the logarithm gives us:[= log_{10}10]Since log_{10}10 = 1, we have:[frac{1}{m+1}+frac{1}{n+1} = 1]Therefore, the correct answer is boxed{B}.

🧠:[<think>Okay, so I need to evaluate the integral from 0 to 1 of |x² - 4| dx. Hmm, absolute value integrals can be tricky because you have to consider where the expression inside the absolute value is positive or negative. Let me start by recalling that the absolute value function |a| is equal to a when a ≥ 0 and -a when a < 0. So, I need to figure out where x² - 4 is positive or negative between x = 0 and x = 1.First, let's solve the inequality x² - 4 ≥ 0. That would be when x² ≥ 4, which implies x ≥ 2 or x ≤ -2. But our interval is from 0 to 1, right? So in this interval, x is between 0 and 1. Let's check x = 0: 0² - 4 = -4, which is negative. At x = 1: 1² - 4 = 1 - 4 = -3, still negative. So in the entire interval [0,1], x² - 4 is negative. That means |x² - 4| = -(x² - 4) = 4 - x². So the integral simplifies to the integral from 0 to 1 of (4 - x²) dx. That seems manageable.Wait, let me double-check that. If x² - 4 is always negative in [0,1], then the absolute value would flip the sign. So yes, |x² - 4| = 4 - x² over the entire interval. Therefore, the integral becomes ∫₀¹ (4 - x²) dx.Okay, let's compute that. The integral of 4 dx is 4x, and the integral of x² dx is (x³)/3. So putting that together, the integral is [4x - (x³)/3] evaluated from 0 to 1. Plugging in x = 1: 4(1) - (1³)/3 = 4 - 1/3 = 11/3. Then subtract the value at x = 0, which is 4(0) - (0³)/3 = 0. So the result is 11/3 - 0 = 11/3. Hmm, 11/3 is option A. But wait, let me check the answer choices again. The options are A: 11/3, B:22/3, C:23/3, D:25/3. So 11/3 is an option. But let me verify once more to make sure I didn't make a mistake.Wait, maybe I messed up the integral. Let me recalculate. The integral of 4 is 4x, correct. The integral of x² is x³/3, correct. So 4x - x³/3. At x=1: 4*1 - 1/3 = 4 - 1/3 = 12/3 -1/3=11/3. At x=0: 0 -0=0. So yes, 11/3. So the answer should be A. But why is there an option C:23/3? Maybe I made a mistake in determining where the expression is negative?Wait, let me check again. The function x² -4. When x is in [0,1], x² is between 0 and 1. So x² -4 is between -4 and -3. Therefore, it's always negative. Hence, |x² -4| = 4 -x². So integrating 4 -x² from 0 to1. So 11/3. So answer is A. But let me check the calculation again.Wait, ∫₀¹ 4 dx = 4*(1 -0) =4. ∫₀¹ x² dx = [x³/3]₀¹ = (1 -0)/3=1/3. Therefore, 4 -1/3=11/3. Yep. So answer is A. Maybe the options are trying to trick me? Or did I misinterpret the question? Wait, the integral is from 0 to1. The integrand is |x² -4|. Hmm, no, seems straightforward.Alternatively, maybe the problem was meant to be from 0 to 3? Because sometimes misread limits can happen. But the user wrote from 0 to1. Let me check again. The original problem is ∫₀¹ |x² -4| dx. So yes, 0 to1. Then the answer should be 11/3, which is option A. But let's look at the options again. The user provided options: A:11/3, B:22/3, C:23/3, D:25/3.Hmm, maybe there's a mistake in my reasoning? Let me think. Wait, is there a possibility that x² -4 changes sign within [0,1]? Let's see. The roots of x² -4 =0 are x=2 and x=-2. Since our interval is from 0 to1, which is between -2 and2, so x² -4 is negative throughout. So |x² -4| is 4 -x². So integrating that from0 to1 gives 11/3. Therefore, answer A.But why does the option C:23/3 exist? Maybe I made a miscalculation. Let me compute the integral step by step again. ∫₀¹ (4 -x²) dx. The integral of 4 is 4x, evaluated from0 to1 gives 4*1 -4*0=4. The integral of x² is x³/3, evaluated from0 to1 gives 1/3 -0=1/3. So subtracting: 4 -1/3= (12/3 -1/3)=11/3. So that's correct. Therefore, answer A.Alternatively, maybe the problem was written incorrectly? If the integral were from 0 to3, for example, the answer would be different. Let's check that. Suppose it's from0 to3. Then, x² -4=0 when x=2. So in [0,2], x² -4 is negative, and in [2,3], positive. So we would split the integral into [0,2] and [2,3]. But in our problem, it's from0 to1, so only [0,1], and x² -4 is negative throughout. So, no, the answer should be 11/3.Therefore, I think the correct answer is A:11/3. Maybe the other options are for different intervals or different functions? But according to the problem given, it's 0 to1 of |x² -4| dx. So 11/3, which is option A.Final Answerboxed{A}</think>To evaluate the integral (int_{0}^{1} |x^2 - 4| , dx), we first determine where the expression inside the absolute value, (x^2 - 4), is positive or negative. For (x) in the interval ([0, 1]), (x^2) ranges from 0 to 1. Since (x^2 - 4) is always negative in this interval (as (x^2 leq 1 < 4)), we have (|x^2 - 4| = 4 - x^2).Thus, the integral simplifies to:[int_{0}^{1} (4 - x^2) , dx]We compute this integral by breaking it into two parts:1. The integral of 4 with respect to (x) is (4x).2. The integral of (x^2) with respect to (x) is (frac{x^3}{3}).Combining these, we evaluate from 0 to 1:[left[ 4x - frac{x^3}{3} right]_{0}^{1}]Substituting the limits of integration:- At (x = 1): (4(1) - frac{1^3}{3} = 4 - frac{1}{3} = frac{12}{3} - frac{1}{3} = frac{11}{3})- At (x = 0): (4(0) - frac{0^3}{3} = 0)Subtracting these results, we get:[frac{11}{3} - 0 = frac{11}{3}]Thus, the value of the integral is (boxed{A}).,<think>Okay, so I need to evaluate the integral from 0 to 1 of the absolute value of x squared minus 4 dx. Hmm, let me think. The integrand is |x² - 4|. Since it's an absolute value, I have to consider where the expression inside the absolute value is positive or negative over the interval [0,1]. First, let's think about x² - 4. For x in [0,1], x² will be between 0 and 1. Then, x² - 4 will be between -4 and -3. So, x² - 4 is always negative on [0,1], right? Because even the maximum x² is 1, so 1 - 4 = -3. So, the absolute value of a negative number is its positive counterpart. Therefore, |x² - 4| = -(x² - 4) = 4 - x² over the entire interval [0,1].That simplifies the integral. So instead of dealing with the absolute value, which can complicate things by splitting the integral into different intervals, here it's straightforward because the expression inside is always negative. So the integral becomes the integral from 0 to 1 of (4 - x²) dx.Alright, let's compute that. The integral of 4 dx is 4x, and the integral of x² dx is (x³)/3. So putting those together, the integral from 0 to 1 of (4 - x²) dx is [4x - (x³)/3] evaluated from 0 to 1.Plugging in the upper limit, x = 1: 4*(1) - (1³)/3 = 4 - 1/3. Then subtracting the lower limit, x = 0: 4*(0) - (0³)/3 = 0. So the entire integral is (4 - 1/3) - 0 = 4 - 1/3.Now, 4 is equal to 12/3, so 12/3 - 1/3 = 11/3. Therefore, the value of the integral is 11/3. Let me check the answer choices: A is 11/3, B is 22/3, C is 23/3, D is 25/3. So the answer must be A: 11/3.Wait, but let me double-check. Did I handle the absolute value correctly? Since x² - 4 is negative on [0,1], then absolute value makes it 4 - x². Integrating that from 0 to 1. The integral of 4 is 4x, evaluated from 0 to1 is 4*1 -4*0 =4. Integral of x² is x³/3, evaluated from 0 to1 is 1/3 -0 =1/3. So 4 -1/3 = 11/3. Yes, that's correct. So the answer is A.But let me just confirm once more. Suppose someone might think that x² -4 crosses zero in [0,1], but x² is at most 1, so x² -4 is at most -3. So no, it doesn't cross zero here. So absolute value is just flipping the sign everywhere in this interval. Therefore, integrating 4 -x² is correct. Then the integral is 11/3. So answer A. I think that's solid. I don't see any mistakes here.Final Answerboxed{A}</think>To evaluate the integral (int_{0}^{1} |x^2 - 4| , dx), we first consider the expression inside the absolute value, (x^2 - 4). For (x) in the interval ([0, 1]), (x^2) ranges from 0 to 1. Therefore, (x^2 - 4) ranges from (-4) to (-3), which is always negative. Consequently, (|x^2 - 4| = -(x^2 - 4) = 4 - x^2) over the entire interval ([0, 1]).The integral simplifies to:[int_{0}^{1} (4 - x^2) , dx]We compute this integral by finding the antiderivative of (4 - x^2):[int (4 - x^2) , dx = 4x - frac{x^3}{3} + C]Evaluating this antiderivative from 0 to 1:[left[ 4x - frac{x^3}{3} right]_{0}^{1} = left( 4(1) - frac{1^3}{3} right) - left( 4(0) - frac{0^3}{3} right) = left( 4 - frac{1}{3} right) - 0 = frac{12}{3} - frac{1}{3} = frac{11}{3}]Thus, the value of the integral is (boxed{A}).]

🔑:Since for 0 leqslant x leqslant 1, we have |x^{2}-4|=4-x^{2},Therefore, int_{0}^{1}|x^{2}-4|dx= int_{0}^{1}(4-x^{2})dx=(4x- dfrac{1}{3}x^{3}) |_{0}^{1}=4- dfrac{1}{3}=dfrac{11}{3}.Hence, the correct choice is boxed{A}.This problem mainly tests the calculation of function integrals, requiring proficiency in the integral formulas of common functions.

🧠:[<think>Okay, so I need to solve this problem where y is the sum of y₁ and y₂. y₁ is directly proportional to x - 3, and y₂ is directly proportional to x² + 1. Then they give me two points: when x=0, y=-2, and when x=1, y=4. I need to find the functional relationship between y and x, and then find where it intersects the coordinate axes.First, let me recall what "directly proportional" means. If something is directly proportional, that means it's a linear relationship with a constant of proportionality. So for y₁, since it's directly proportional to x - 3, I can write that as y₁ = k(x - 3), where k is the constant of proportionality. Similarly, y₂ is directly proportional to x² + 1, so y₂ = m(x² + 1), where m is another constant. Since y is the sum of y₁ and y₂, then y = k(x - 3) + m(x² + 1). So my goal is to find the constants k and m. To do that, I can use the given points. When x=0, y=-2, and when x=1, y=4. Let me plug those into the equation and set up a system of equations to solve for k and m.Starting with x=0. Plugging x=0 into the equation:y = k(0 - 3) + m(0² + 1) = k(-3) + m(1) = -3k + mAnd they told me that when x=0, y=-2. So:-3k + m = -2 ...(1)Now, plugging in x=1:y = k(1 - 3) + m(1² + 1) = k(-2) + m(2) = -2k + 2mAnd they said when x=1, y=4. So:-2k + 2m = 4 ...(2)Now, we have two equations:1) -3k + m = -22) -2k + 2m = 4I need to solve this system for k and m. Let me try solving equation (1) for m first. From equation (1):m = 3k - 2Then substitute this into equation (2):-2k + 2(3k - 2) = 4Let me compute that:-2k + 6k - 4 = 4Combine like terms:4k - 4 = 4Add 4 to both sides:4k = 8Divide both sides by 4:k = 2Then substitute back into m = 3k - 2:m = 3*2 - 2 = 6 - 2 = 4So k=2 and m=4. Therefore, the functional relationship is:y = 2(x - 3) + 4(x² + 1)Let me expand that to make it a standard quadratic equation.First, distribute the 2 and the 4:2x - 6 + 4x² + 4Combine like terms:4x² + 2x - 6 + 4 = 4x² + 2x - 2So the equation is y = 4x² + 2x - 2.Let me check if this works with the given points.For x=0: y = 0 + 0 - 2 = -2. Correct.For x=1: y = 4(1) + 2(1) - 2 = 4 + 2 - 2 = 4. Correct.Good, so that seems to check out. So part (1) is done. The functional relationship is y = 4x² + 2x - 2.Now, part (2): Find the coordinates of the intersection points of the function with the coordinate axes.Coordinate axes are the x-axis and y-axis. So intersections with the x-axis are the points where y=0, and intersection with the y-axis is where x=0. They might be asking for both.We already know that when x=0, y=-2. So the intersection with the y-axis is (0, -2). So that's one point.For the x-axis intersections, set y=0 and solve for x:4x² + 2x - 2 = 0Let me solve this quadratic equation. Let's write it as:4x² + 2x - 2 = 0I can simplify this equation by dividing all terms by 2:2x² + x - 1 = 0Now, let's apply the quadratic formula. For ax² + bx + c = 0, the solutions are:x = [-b ± sqrt(b² - 4ac)]/(2a)Here, a=2, b=1, c=-1.So discriminant D = b² - 4ac = 1² - 4*2*(-1) = 1 + 8 = 9Square root of D is 3.Thus,x = [-1 ± 3]/(2*2) = (-1 ± 3)/4So two solutions:1) (-1 + 3)/4 = 2/4 = 1/22) (-1 - 3)/4 = -4/4 = -1Therefore, the x-intercepts are at x=1/2 and x=-1. So the coordinates are (1/2, 0) and (-1, 0).Let me verify these solutions. Plugging x=1/2 into the original equation:y = 4*(1/2)^2 + 2*(1/2) - 2 = 4*(1/4) + 1 - 2 = 1 + 1 - 2 = 0. Correct.For x=-1:y = 4*(-1)^2 + 2*(-1) - 2 = 4*1 - 2 - 2 = 4 - 4 = 0. Correct.So the intersection points with the coordinate axes are (0, -2), (1/2, 0), and (-1, 0).But wait, the problem says "coordinates of the intersection points of the function with the coordinate axes". So that would include both x-intercepts and y-intercept. So all three points.But maybe in the problem, "coordinate axes" refers to each axis, so x-axis and y-axis. So the intersections with the x-axis are two points: (-1, 0) and (1/2, 0), and intersection with the y-axis is (0, -2). So all three points.But let me check the problem statement again. It says "the coordinates of the intersection points of the function with the coordinate axes." So maybe they want all the points where the graph intersects the axes, which are three points. So the answer for part (2) would be those three points.But let me check if that's the case. The function is a quadratic, so it can intersect the x-axis at two points and the y-axis at one point. So yes, three points in total.But wait, wait. Wait, actually, for a quadratic function, the maximum number of x-intercepts is two, and one y-intercept. So that's correct.Therefore, the intersection points are (0, -2), (-1, 0), and (1/2, 0).So summarizing:(1) The functional relationship is y = 4x² + 2x - 2.(2) The intersection points are (0, -2), (-1, 0), and (1/2, 0).But let me check again if that's correct.First, part (1) we derived y = 4x² + 2x - 2, which matches the given points. So that's correct.For part (2), solving 4x² + 2x - 2 = 0 gives x = 1/2 and x = -1, so the x-intercepts are (-1, 0) and (1/2, 0), and the y-intercept is (0, -2). So all correct.Therefore, these should be the answers.Final Answer(1) The functional relationship is boxed{y = 4x^2 + 2x - 2}.(2) The coordinates of the intersection points are boxed{(-1, 0)}, boxed{left(dfrac{1}{2}, 0right)}, and boxed{(0, -2)}.</think>Given ( y = y_1 + y_2 ), where ( y_1 ) is directly proportional to ( x - 3 ), and ( y_2 ) is directly proportional to ( x^2 + 1 ). We know that when ( x = 0 ), ( y = -2 ), and when ( x = 1 ), ( y = 4 ).(1) To find the functional relationship between ( y ) and ( x ):Since ( y_1 ) is directly proportional to ( x - 3 ), we write ( y_1 = k(x - 3) ). Similarly, ( y_2 ) is directly proportional to ( x^2 + 1 ), so ( y_2 = m(x^2 + 1) ). Therefore, ( y = k(x - 3) + m(x^2 + 1) ).Using the given points:- When ( x = 0 ), ( y = -2 ): [ -3k + m = -2 quad text{(1)} ]- When ( x = 1 ), ( y = 4 ): [ -2k + 2m = 4 quad text{(2)} ]Solving the system of equations:From equation (1), ( m = 3k - 2 ).Substitute ( m ) into equation (2):[-2k + 2(3k - 2) = 4 -2k + 6k - 4 = 4 4k - 4 = 4 4k = 8 k = 2]Then, ( m = 3(2) - 2 = 4 ).Thus, the functional relationship is:[y = 2(x - 3) + 4(x^2 + 1) = 4x^2 + 2x - 2](2) To find the coordinates of the intersection points with the coordinate axes:- Y-intercept: When ( x = 0 ), ( y = -2 ), so the point is ( (0, -2) ).- X-intercepts: Set ( y = 0 ): [ 4x^2 + 2x - 2 = 0 ] Dividing by 2: [ 2x^2 + x - 1 = 0 ] Using the quadratic formula ( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ) with ( a = 2 ), ( b = 1 ), ( c = -1 ): [ x = frac{-1 pm sqrt{1 + 8}}{4} = frac{-1 pm 3}{4} ] Solutions: [ x = frac{2}{4} = frac{1}{2} quad text{and} quad x = frac{-4}{4} = -1 ] Thus, the x-intercepts are ( left( frac{1}{2}, 0 right) ) and ( (-1, 0) ).Final answers:(1) The functional relationship is boxed{y = 4x^2 + 2x - 2}.(2) The coordinates of the intersection points are boxed{(-1, 0)}, boxed{left(dfrac{1}{2}, 0right)}, and boxed{(0, -2)}.,<think>Okay, let's see. I need to find the functional relationship between y and x. The problem says that y is the sum of y1 and y2, where y1 is directly proportional to x - 3, and y2 is directly proportional to x squared plus 1. Then there are two specific points given: when x is 0, y is -2, and when x is 1, y is 4. First, let me recall what "directly proportional" means. If a variable, say y1, is directly proportional to another variable, say (x - 3), that means there's a constant of proportionality, let's call it k1, such that y1 = k1*(x - 3). Similarly, y2 would be y2 = k2*(x² + 1). So, the total y would be y = y1 + y2 = k1*(x - 3) + k2*(x² + 1). So the equation is y = k1*(x - 3) + k2*(x² + 1). My job is to find the constants k1 and k2. To do that, I can use the two given points. When x = 0, y = -2, and when x = 1, y = 4. Let me plug in x = 0 first. Substituting into the equation:-2 = k1*(0 - 3) + k2*(0 + 1) -2 = k1*(-3) + k2*(1) So that simplifies to -3*k1 + k2 = -2. Let me write that as equation (1): -3k1 + k2 = -2.Now plug in x = 1. Then:4 = k1*(1 - 3) + k2*(1² + 1) 4 = k1*(-2) + k2*(2) So that's -2*k1 + 2*k2 = 4. Let's call this equation (2): -2k1 + 2k2 = 4.Now I need to solve these two equations for k1 and k2. Let me write them again:Equation (1): -3k1 + k2 = -2 Equation (2): -2k1 + 2k2 = 4Hmm. Maybe I can use substitution or elimination. Let's try elimination. If I multiply equation (1) by 2, then I can eliminate k2 or k1. Let's see:Multiply equation (1) by 2: -6k1 + 2k2 = -4Now subtract equation (2) from this result: (-6k1 + 2k2) - (-2k1 + 2k2) = -4 - 4 Wait, no, the right-hand side should be -4 - 4? Wait, original equations:Equation (1) multiplied by 2: -6k1 + 2k2 = -4 Equation (2): -2k1 + 2k2 = 4If I subtract equation (2) from the multiplied equation (1), then:(-6k1 + 2k2) - (-2k1 + 2k2) = -4 - 4 Calculating left side: -6k1 + 2k2 + 2k1 - 2k2 = (-6k1 + 2k1) + (2k2 - 2k2) = -4k1 + 0 = -4k1 Right side: -4 - 4 = -8 So -4k1 = -8 Divide both sides by -4: k1 = (-8)/(-4) = 2So k1 is 2. Then plug back into equation (1) to find k2.Equation (1): -3k1 + k2 = -2 Substitute k1 = 2: -3*(2) + k2 = -2 -6 + k2 = -2 Add 6 to both sides: k2 = -2 + 6 = 4So k2 is 4. Therefore, the functional relationship is:y = 2*(x - 3) + 4*(x² + 1)Let me expand that to write it in standard form. First distribute the 2 and 4:y = 2x - 6 + 4x² + 4 Combine like terms: 4x² + 2x + (-6 + 4) = 4x² + 2x - 2So, the equation is y = 4x² + 2x - 2. Let me check if this satisfies the given points.When x = 0: y = 4*(0)^2 + 2*0 -2 = -2. Correct. When x = 1: y = 4*(1)^2 + 2*1 -2 = 4 + 2 -2 = 4. Correct. So that's part (1) done.Now part (2): Find the coordinates of the intersection points of the function with the coordinate axes. The coordinate axes are the x-axis and y-axis. Intersection with the y-axis occurs when x = 0. We already know that when x = 0, y = -2. So the point is (0, -2).Intersection with the x-axis occurs when y = 0. So we need to solve 4x² + 2x - 2 = 0. Let me solve this quadratic equation.The quadratic equation is 4x² + 2x - 2 = 0. Let's use the quadratic formula:x = [-b ± sqrt(b² - 4ac)]/(2a)Here, a = 4, b = 2, c = -2.Discriminant D = b² - 4ac = (2)^2 - 4*4*(-2) = 4 + 32 = 36Since D is 36, sqrt(36) = 6. So,x = [-2 ± 6]/(2*4) = (-2 ±6)/8So two solutions:First solution: (-2 + 6)/8 = 4/8 = 1/2 Second solution: (-2 -6)/8 = -8/8 = -1Therefore, the x-intercepts are at x = 1/2 and x = -1. Therefore, the coordinates are (1/2, 0) and (-1, 0).Wait, let me check if these are correct by plugging back into the equation.For x = 1/2:y = 4*(1/2)^2 + 2*(1/2) -2 = 4*(1/4) + 1 -2 = 1 +1 -2 = 0. Correct.For x = -1:y = 4*(-1)^2 + 2*(-1) -2 = 4*1 -2 -2 = 4 -4 = 0. Correct.So the intersection points are (0, -2), (1/2, 0), and (-1, 0). Wait, but the problem says "the coordinates of the intersection points of the function with the coordinate axes." So coordinate axes are x-axis and y-axis. So the intersections with the y-axis is (0, -2), and with the x-axis are (-1, 0) and (1/2, 0). So all three points.Therefore, part (2) is done.Wait, but the problem mentions "coordinates of the intersection points", so maybe list all three points? But in the problem statement, when they say "the function with the coordinate axes", I think it's the intersections with both axes. So the y-intercept is one point, and the x-intercepts are two points. So all three points. So answer (0, -2), (-1, 0), and (1/2, 0). But let me check again.Yes, correct.So summarizing:1) The functional relationship is y = 4x² + 2x -2.2) The intersection points are (0, -2), (-1, 0), and (1/2, 0).Wait, but the problem says "the coordinates of the intersection points of the function with the coordinate axes." So coordinate axes are x-axis and y-axis, so all intercepts. So indeed, three points: one y-intercept and two x-intercepts.But let me make sure that part (2) is only asking for the intersection with the coordinate axes, which includes both x and y axes. So all intercepts. So the answer for part (2) should list all those points.Therefore, the final answers are:1) y = 4x² + 2x - 22) Intersection points: (0, -2), (-1, 0), and (1/2, 0)Wait, but in the problem, part (1) is to find the functional relationship, which I did. Part (2) is to find the coordinates of the intersection points with the coordinate axes, which I did. So that's all.But let me check again if I made any calculation errors. Let's go through the steps again.Starting with y = y1 + y2, where y1 is proportional to x - 3, and y2 proportional to x² + 1. So y = k1(x - 3) + k2(x² +1). Then using x=0, y=-2:At x=0: y = k1*(-3) + k2*(1) = -3k1 + k2 = -2. At x=1: y = k1*(-2) + k2*(2) = -2k1 + 2k2 = 4.Solving those equations:Equation 1: -3k1 + k2 = -2 Equation 2: -2k1 + 2k2 = 4Multiply equation 1 by 2: -6k1 + 2k2 = -4 Subtract equation 2: (-6k1 + 2k2) - (-2k1 + 2k2) = -4 -4 Wait, no. Wait, equation 2 is -2k1 + 2k2 =4. So if we subtract equation 2 from the multiplied equation 1, which is:Left side: (-6k1 + 2k2) - (-2k1 + 2k2) = (-6k1 + 2k2) + 2k1 -2k2 = (-4k1) Right side: -4 -4 = -8. So -4k1 = -8, so k1=2. Correct. Then k2 = -2 +3k1 = -2 +6=4. Correct.So substituting back, y = 2(x -3) +4(x² +1). Expand: 2x -6 +4x² +4 =4x² +2x -2. Correct. Then intercepts: y-intercept at (0,-2). X-intercepts solving 4x² +2x -2=0. Quadratic formula: x=(-2 ±sqrt(4 +32))/8=(-2±6)/8, so 4/8=0.5, -8/8=-1. Correct.So all steps check out. So the answers are correct.Final Answer(1) The functional relationship is boxed{y = 4x^2 + 2x - 2}. (2) The coordinates of the intersection points are boxed{(0, -2)}, boxed{(-1, 0)}, and boxed{left(dfrac{1}{2}, 0right)}.</think>Given ( y = y_1 + y_2 ), where ( y_1 ) is directly proportional to ( x - 3 ), and ( y_2 ) is directly proportional to ( x^2 + 1 ). When ( x = 0 ), ( y = -2 ), and when ( x = 1 ), ( y = 4 ).(1) To find the functional relationship between ( y ) and ( x ):We start by expressing ( y_1 ) and ( y_2 ) with constants of proportionality ( k_1 ) and ( k_2 ):[ y_1 = k_1 (x - 3) ][ y_2 = k_2 (x^2 + 1) ]Thus, the total ( y ) is:[ y = k_1 (x - 3) + k_2 (x^2 + 1) ]Using the given points:- When ( x = 0 ), ( y = -2 ): [ -2 = k_1 (0 - 3) + k_2 (0 + 1) ] [ -2 = -3k_1 + k_2 ] This gives the equation: ( -3k_1 + k_2 = -2 ).- When ( x = 1 ), ( y = 4 ): [ 4 = k_1 (1 - 3) + k_2 (1^2 + 1) ] [ 4 = -2k_1 + 2k_2 ] This gives the equation: ( -2k_1 + 2k_2 = 4 ).Solving these equations:1. Multiply the first equation by 2: ( -6k_1 + 2k_2 = -4 )2. Subtract the second equation from this result: [ (-6k_1 + 2k_2) - (-2k_1 + 2k_2) = -4 - 4 ] [ -4k_1 = -8 ] [ k_1 = 2 ]Substitute ( k_1 = 2 ) back into the first equation:[ -3(2) + k_2 = -2 ][ -6 + k_2 = -2 ][ k_2 = 4 ]Thus, the functional relationship is:[ y = 2(x - 3) + 4(x^2 + 1) ]Expanding and combining like terms:[ y = 4x^2 + 2x - 2 ](2) To find the coordinates of the intersection points with the coordinate axes:- Y-intercept: When ( x = 0 ), ( y = -2 ). The point is ( (0, -2) ).- X-intercepts: Solve ( 4x^2 + 2x - 2 = 0 ) using the quadratic formula: [ x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ] Here, ( a = 4 ), ( b = 2 ), ( c = -2 ): [ x = frac{-2 pm sqrt{4 + 32}}{8} ] [ x = frac{-2 pm 6}{8} ] [ x = frac{4}{8} = frac{1}{2} quad text{or} quad x = frac{-8}{8} = -1 ]The coordinates of the intersection points are:- Y-intercept: ( (0, -2) )- X-intercepts: ( left( frac{1}{2}, 0 right) ) and ( (-1, 0) ) Final Answer(1) The functional relationship is boxed{y = 4x^2 + 2x - 2}. (2) The coordinates of the intersection points are boxed{(0, -2)}, boxed{(-1, 0)}, and boxed{left( dfrac{1}{2}, 0 right)}.]

🔑: Problem Solution:# Part 1: Finding the Functional RelationshipGiven the information, we can express y_{1} and y_{2} as follows:- y_{1}=k_{1}(x-3), since y_{1} is directly proportional to x-3.- y_{2}=k_{2}(x^{2}+1), since y_{2} is directly proportional to x^{2}+1.Thus, the total y can be expressed as a combination of y_{1} and y_{2}:y = y_{1} + y_{2} = k_{1}(x-3) + k_{2}(x^{2}+1)Substituting the given points (x=0, y=-2) and (x=1, y=4) into the equation, we obtain a system of equations:begin{cases}-3k_{1} + k_{2} = -2 -2k_{1} + 2k_{2} = 4end{cases}Solving this system, we find:k_{1} = 2, quad k_{2} = 4Therefore, the functional relationship between y and x is:y = 4x^{2} + 2x - 2boxed{y = 4x^{2} + 2x - 2}# Part 2: Finding the Intersection Points with Coordinate Axes- Intersection with the Y-axis (x=0):Substituting x=0 into the equation y=4x^{2}+2x-2, we get:y = 4(0)^{2} + 2(0) - 2 = -2Therefore, the intersection point with the y-axis is:boxed{(0, -2)}- Intersection with the X-axis (y=0):Setting y=0 in the equation y=4x^{2}+2x-2, we solve for x:4x^{2} + 2x - 2 = 0Factoring or using the quadratic formula, we find:x_{1} = -1, quad x_{2} = frac{1}{2}Therefore, the intersection points with the x-axis are:boxed{(-1, 0)} quad text{and} quad boxed{left(frac{1}{2}, 0right)} Analysis:- The method of undetermined coefficients was effectively used to establish the relationship between y and x.- Substituting specific values for x and y allowed us to find the intersection points with the coordinate axes. Comments:This problem demonstrates the utility of the method of undetermined coefficients in finding the equation of a function given specific conditions. It also illustrates how to find intersection points with the coordinate axes by substituting appropriate values into the equation.

🧠:[<think>Okay, let's try to tackle this problem step by step. The question says that for a complex number z = x + yi, the ratio of the real part to the imaginary part of (z + 1)/(z + 2) is √3. We need to find the range of y/x. Hmm, sounds a bit tricky, but let me break it down.First, let's write down what (z + 1)/(z + 2) is in terms of x and y. Since z = x + yi, then z + 1 = (x + 1) + yi, and z + 2 = (x + 2) + yi. So, we have to compute the division of these two complex numbers.To divide two complex numbers, we can multiply the numerator and denominator by the conjugate of the denominator. The conjugate of (x + 2 + yi) is (x + 2 - yi). Let's do that:[( (x + 1) + yi ) * ( (x + 2) - yi )] / [ ( (x + 2) + yi ) * ( (x + 2) - yi ) ]First, compute the denominator:Denominator = (x + 2)^2 + y^2. That's straightforward because when you multiply a complex number by its conjugate, you get the square of the real part plus the square of the imaginary part.Now, the numerator is a bit more involved. Let's expand that:Numerator = (x + 1)(x + 2) - (x + 1)yi + yi(x + 2) - yi * yiWait, let's do this step by step. Multiplying (x + 1 + yi)(x + 2 - yi):First, multiply (x + 1)(x + 2) = x^2 + 3x + 2.Then, (x + 1)(-yi) = -y(x + 1)i.Then, yi*(x + 2) = y(x + 2)i.Finally, yi*(-yi) = -y^2 i^2 = -y^2*(-1) = y^2.So combining all these terms:Real parts: x^2 + 3x + 2 + y^2.Imaginary parts: [-y(x + 1) + y(x + 2)]i.Simplify the imaginary part:-y(x + 1) + y(x + 2) = -yx - y + yx + 2y = (-yx + yx) + (-y + 2y) = y.Therefore, the numerator is (x^2 + 3x + 2 + y^2) + y i.So the expression (z + 1)/(z + 2) is equal to [ (x^2 + 3x + 2 + y^2) + y i ] / [ (x + 2)^2 + y^2 ].Since both numerator and denominator are real numbers except for the imaginary part, we can separate the real and imaginary parts:Real part: (x^2 + 3x + 2 + y^2) / [ (x + 2)^2 + y^2 ]Imaginary part: y / [ (x + 2)^2 + y^2 ]The problem states that the ratio of the real part to the imaginary part is √3. So, we can set up the equation:Real Part / Imaginary Part = √3So:[ (x^2 + 3x + 2 + y^2 ) / D ] / [ y / D ] = √3, where D is the denominator, (x + 2)^2 + y^2. Since D is non-zero (unless z = -2, but z can't be -2 because we are dividing by z + 2), so we can safely cancel D:( x^2 + 3x + 2 + y^2 ) / y = √3Therefore:x^2 + 3x + 2 + y^2 = √3 yHmm, okay. Now, we need to find the range of y/x. Let's denote k = y/x. Then y = kx. Let's substitute this into the equation.So substituting y = kx into the equation:x^2 + 3x + 2 + (k x)^2 = √3 * k xLet's rearrange terms:x^2 + 3x + 2 + k^2 x^2 - √3 k x = 0Combine like terms:(1 + k^2) x^2 + (3 - √3 k) x + 2 = 0Now, this is a quadratic equation in x. For real solutions x to exist, the discriminant of this quadratic must be non-negative.Discriminant D = [ (3 - √3 k)^2 - 4 * (1 + k^2) * 2 ] ≥ 0Let's compute this discriminant:First, expand (3 - √3 k)^2:= 9 - 6√3 k + 3k^2Then compute 4*(1 + k^2)*2:= 8*(1 + k^2) = 8 + 8k^2So discriminant D = (9 - 6√3 k + 3k^2) - (8 + 8k^2) ≥ 0Simplify:9 - 6√3 k + 3k^2 - 8 - 8k^2 ≥ 0Combine like terms:(9 - 8) + (-6√3 k) + (3k^2 - 8k^2) ≥ 01 - 6√3 k -5k^2 ≥ 0Multiply both sides by -1 (which reverses the inequality):5k^2 + 6√3 k -1 ≤ 0So we have the quadratic inequality 5k^2 + 6√3 k -1 ≤ 0.To find the values of k that satisfy this inequality, we can first find the roots of the quadratic equation 5k^2 + 6√3 k -1 = 0.Using the quadratic formula:k = [ -6√3 ± √( (6√3)^2 - 4*5*(-1) ) ] / (2*5)Compute discriminant inside the square root:(6√3)^2 - 4*5*(-1) = 36*3 + 20 = 108 + 20 = 128So sqrt(128) = 8√2Therefore, k = [ -6√3 ± 8√2 ] / 10So the two roots are:k1 = [ -6√3 + 8√2 ] / 10k2 = [ -6√3 - 8√2 ] / 10Since the quadratic coefficient (5) is positive, the quadratic opens upwards. Therefore, the inequality 5k^2 + 6√3 k -1 ≤ 0 is satisfied between the two roots.Hence, the range of k = y/x is [ k2, k1 ], i.e., from [ (-6√3 - 8√2 ) / 10 , ( -6√3 + 8√2 ) / 10 ]But let's verify if this is correct.Wait, but maybe I made a miscalculation in the discriminant step. Let's check again.Original discriminant calculation:D = (3 - √3 k)^2 - 8*(1 + k^2)Expanding (3 - √3 k)^2: 9 - 6√3 k + 3k^2Then subtract 8*(1 + k^2): 9 -6√3 k +3k^2 -8 -8k^2 = 1 -6√3 k -5k^2. That's correct.Then multiplied by -1: 5k^2 +6√3 k -1 ≤0, correct.Quadratic equation: 5k^2 +6√3 k -1 =0Discriminant: (6√3)^2 -4*5*(-1) = 108 +20=128, which is 128. Square root of 128 is 8√2. Correct.Hence, roots:[ -6√3 ±8√2 ] /10So, yes, the roots are as calculated.Therefore, the range of k is [ ( -6√3 -8√2 )/10 , ( -6√3 +8√2 )/10 ]But we can write this in a simplified form:Factor numerator:For the upper bound: ( -6√3 +8√2 ) /10 = (8√2 -6√3)/10 = (4√2 -3√3)/5Similarly, lower bound: ( -6√3 -8√2 ) /10 = ( -6√3 -8√2 ) /10 = -(6√3 +8√2)/10 = -(3√3 +4√2)/5Therefore, the range is [ - (3√3 +4√2)/5 , (4√2 -3√3)/5 ]But let's check if this is the correct answer.Wait, but the problem says "the range of y/x". So k = y/x is between these two values.Alternatively, maybe there's a different way to approach this problem, perhaps geometrically?Alternatively, we can consider that z is a point (x, y) in the complex plane, and (z +1)/(z +2) is another complex number whose real part over imaginary part is √3. Let's see.Let me think.Suppose we denote w = (z +1)/(z +2). Then the ratio of the real part to the imaginary part of w is √3. So Re(w)/Im(w) = √3. That is, Re(w) = √3 Im(w). So, w lies on the line in the complex plane where the real part is √3 times the imaginary part. So, the argument of w is arctangent of (Im(w)/Re(w)) = arctangent(1/√3) = 30 degrees. Wait, but if Re(w) = √3 Im(w), then tan(theta) = Im(w)/Re(w) = 1/√3, so theta = 30 degrees. So w lies on the line making 30 degrees with the positive real axis.Alternatively, since Re(w)/Im(w) = √3, so if we write w = a + b i, then a/b = √3, so a = √3 b. So, the complex number w lies on the line a = √3 b, except when b =0. But in our case, since (z +1)/(z +2) is defined (z ≠ -2), and the ratio is given as √3, so the imaginary part of w can't be zero. Therefore, z cannot be such that (z +1)/(z +2) is real. So, perhaps z lies on some circle or line?Alternatively, maybe we can consider the transformation w = (z +1)/(z +2). This is a Möbius transformation. The image of the real line under this transformation would be a circle or a line. However, since we are dealing with the condition that w lies on the line Re(w)/Im(w) = √3, perhaps we can find the locus of z such that w lies on that line.But maybe this approach is more complicated. Let's stick with the algebra we did earlier.So, from our previous steps, we arrived at the quadratic inequality in k:5k^2 +6√3 k -1 ≤0Which gives us the range of k between the roots. Therefore, the answer should be the interval [ ( -6√3 -8√2 )/10, ( -6√3 +8√2 )/10 ].But let's compute numerical values to see what this is approximately, to check if it makes sense.Compute 3√3 ≈ 3*1.732 ≈5.196, 4√2≈4*1.414≈5.656So, for the upper bound: (4√2 -3√3)/5 ≈(5.656 -5.196)/5≈0.46/5≈0.092Lower bound: -(3√3 +4√2)/5≈-(5.196 +5.656)/5≈-10.852/5≈-2.170So the range of y/x is from approximately -2.17 to 0.092. Hmmm, so negative values and a small positive value.Wait, but is this correct? Let's see. Let me think if there might be a mistake in the algebra.Wait, when we substituted y = kx into the equation x^2 +3x +2 + y^2 = √3 y, we get x^2 +3x +2 +k²x² =√3 k x, which is correct.Then rearranged to (1 +k²)x² + (3 -√3 k)x +2=0, correct.Then discriminant D=(3 -√3 k)^2 -4*(1 +k²)*2 ≥0.Which is 9 -6√3 k +3k² -8 -8k² ≥0 →1 -6√3 k -5k² ≥0. Yes, correct.Multiply by -1: 5k² +6√3 k -1 ≤0. Correct.So quadratic equation 5k² +6√3 k -1 =0, discriminant 128, roots as computed.Therefore, seems correct.Therefore, the range is [ (-6√3 -8√2)/10, (-6√3 +8√2)/10 ]But to write this in terms of radicals:First root: (-6√3 -8√2)/10 = - (6√3 +8√2)/10 = - (3√3 +4√2)/5Second root: (-6√3 +8√2)/10 = (8√2 -6√3)/10 = (4√2 -3√3)/5Therefore, the range is [ - (3√3 +4√2)/5 , (4√2 -3√3)/5 ]Hence, this is the answer.But maybe the question expects the answer in terms of interval notation with radicals simplified, as above.Alternatively, the problem might have another approach leading to a different form, but given the steps I took seem correct, I think this is the right answer.Therefore, the range of y/x is from - (3√3 +4√2)/5 to (4√2 -3√3)/5.Final AnswerThe range of (frac{y}{x}) is (boxed{left[ -dfrac{3sqrt{3} + 4sqrt{2}}{5}, dfrac{4sqrt{2} - 3sqrt{3}}{5} right]}).</think>Given the complex number ( z = x + yi ) where ( x, y in mathbb{R} ), we need to find the range of ( frac{y}{x} ) given that the ratio of the real part to the imaginary part of ( frac{z+1}{z+2} ) is ( sqrt{3} ).First, we express ( frac{z+1}{z+2} ) in terms of ( x ) and ( y ). Let ( z + 1 = (x + 1) + yi ) and ( z + 2 = (x + 2) + yi ). We multiply the numerator and denominator by the conjugate of the denominator:[frac{(x + 1 + yi)(x + 2 - yi)}{(x + 2)^2 + y^2}]Expanding the numerator, we separate the real and imaginary parts:- Real part: ( x^2 + 3x + 2 + y^2 )- Imaginary part: ( y )Thus, the expression simplifies to:[frac{x^2 + 3x + 2 + y^2 + yi}{(x + 2)^2 + y^2}]Given the ratio of the real part to the imaginary part is ( sqrt{3} ):[frac{x^2 + 3x + 2 + y^2}{y} = sqrt{3}]This simplifies to:[x^2 + 3x + 2 + y^2 = sqrt{3} y]Substituting ( y = kx ) (where ( k = frac{y}{x} )) into the equation:[x^2 + 3x + 2 + (kx)^2 = sqrt{3} kx]Rearranging terms, we get a quadratic equation in ( x ):[(1 + k^2)x^2 + (3 - sqrt{3}k)x + 2 = 0]For real solutions in ( x ), the discriminant must be non-negative:[(3 - sqrt{3}k)^2 - 4(1 + k^2) cdot 2 geq 0]Simplifying the discriminant:[9 - 6sqrt{3}k + 3k^2 - 8 - 8k^2 geq 0 implies 1 - 6sqrt{3}k - 5k^2 geq 0]Multiplying by -1:[5k^2 + 6sqrt{3}k - 1 leq 0]Solving the quadratic inequality, we find the roots using the quadratic formula:[k = frac{-6sqrt{3} pm sqrt{128}}{10} = frac{-6sqrt{3} pm 8sqrt{2}}{10}]Thus, the roots are:[k_1 = frac{4sqrt{2} - 3sqrt{3}}{5}, quad k_2 = frac{-3sqrt{3} - 4sqrt{2}}{5}]The range of ( frac{y}{x} ) is between these roots:[boxed{left[ -dfrac{3sqrt{3} + 4sqrt{2}}{5}, dfrac{4sqrt{2} - 3sqrt{3}}{5} right]}],<think>Okay, so I have this complex number problem here. Let me try to understand it step by step. The problem says:"Let the complex number ( z = x + yi ) satisfy the ratio of the real part to the imaginary part of ( frac{z + 1}{z + 2} ) is ( sqrt{3} ), where ( i ) is the imaginary unit, ( x, y in mathbb{R} ). Then the range of ( frac{y}{x} ) is...?"Alright, so we need to find the possible values of ( frac{y}{x} ) given that the ratio of the real part to the imaginary part of ( frac{z + 1}{z + 2} ) is ( sqrt{3} ). Let me break this down.First, let me recall that for any complex number, when you have a fraction like ( frac{a + bi}{c + di} ), you can write it in terms of its real and imaginary parts by multiplying numerator and denominator by the conjugate of the denominator. Maybe that's the way to go here.Given ( z = x + yi ), then ( z + 1 = (x + 1) + yi ) and ( z + 2 = (x + 2) + yi ). So, the expression ( frac{z + 1}{z + 2} ) would be ( frac{(x + 1) + yi}{(x + 2) + yi} ).Let me denote this expression as ( frac{A}{B} ), where ( A = (x + 1) + yi ) and ( B = (x + 2) + yi ). To separate the real and imaginary parts of ( frac{A}{B} ), I need to multiply numerator and denominator by the conjugate of ( B ), which is ( overline{B} = (x + 2) - yi ).So, ( frac{A}{B} = frac{A cdot overline{B}}{B cdot overline{B}} ).Calculating the denominator first: ( B cdot overline{B} = (x + 2)^2 + y^2 ).Now, the numerator: ( A cdot overline{B} = [(x + 1) + yi][(x + 2) - yi] ).Let me expand that:First, multiply ( (x + 1)(x + 2) ), then ( (x + 1)(-yi) ), then ( yi(x + 2) ), and finally ( yi cdot (-yi) ).So:( (x + 1)(x + 2) = x^2 + 3x + 2 )( (x + 1)(-yi) = -yi(x + 1) = -y(x + 1)i )( yi(x + 2) = y(x + 2)i )( yi(-yi) = -y^2 i^2 = -y^2(-1) = y^2 )Putting all these together:Real parts: ( x^2 + 3x + 2 + y^2 )Imaginary parts: ( [-y(x + 1) + y(x + 2)]i )Simplify the imaginary part:( -y(x + 1) + y(x + 2) = -yx - y + yx + 2y = (-yx + yx) + (-y + 2y) = 0 + y = y )Therefore, the numerator is ( (x^2 + 3x + 2 + y^2) + yi )So, the entire expression ( frac{z + 1}{z + 2} ) is:( frac{x^2 + 3x + 2 + y^2}{(x + 2)^2 + y^2} + frac{y}{(x + 2)^2 + y^2}i )Hence, the real part is ( frac{x^2 + 3x + 2 + y^2}{(x + 2)^2 + y^2} ) and the imaginary part is ( frac{y}{(x + 2)^2 + y^2} ).The problem states that the ratio of the real part to the imaginary part is ( sqrt{3} ). So, we can set up the equation:( frac{text{Real part}}{text{Imaginary part}} = sqrt{3} )Substituting the expressions we found:( frac{frac{x^2 + 3x + 2 + y^2}{(x + 2)^2 + y^2}}{frac{y}{(x + 2)^2 + y^2}} = sqrt{3} )Simplify this ratio:The denominators are the same, so they cancel out:( frac{x^2 + 3x + 2 + y^2}{y} = sqrt{3} )So, we have:( x^2 + 3x + 2 + y^2 = sqrt{3} y )Let me rearrange this equation:( x^2 + y^2 + 3x + 2 - sqrt{3} y = 0 )Hmm, this looks like the equation of a circle. Let me check that.The general equation of a circle is ( x^2 + y^2 + Dx + Ey + F = 0 ). Comparing, we have D = 3, E = -√3, and F = 2.So, the equation represents a circle. To find the range of ( frac{y}{x} ), which is the slope of the line from the origin to the point (x, y) on the circle. So, the problem reduces to finding the range of slopes of lines from the origin to points on this circle.Alternatively, the range of ( frac{y}{x} ) is the set of all possible values of the slope of lines through the origin intersecting the circle. So, geometrically, this is equivalent to finding the tangent lines from the origin to the circle and determining their slopes. The range will be between the minimum and maximum slopes.Let me verify this approach.First, let's write the equation of the circle in standard form.Given the general equation ( x^2 + y^2 + 3x - sqrt{3} y + 2 = 0 ), we can complete the squares.For the x-terms: ( x^2 + 3x ). Completing the square: take half of 3, which is 1.5, square it: ( (3/2)^2 = 9/4 ).For the y-terms: ( y^2 - sqrt{3} y ). Completing the square: take half of -√3, which is -√3/2, square it: ( (√3/2)^2 = 3/4.So, adding and subtracting these:( x^2 + 3x + 9/4 - 9/4 + y^2 - √3 y + 3/4 - 3/4 + 2 = 0 )Which becomes:( (x + 3/2)^2 - 9/4 + (y - √3/2)^2 - 3/4 + 2 = 0 )Combine constants:-9/4 - 3/4 + 2 = (-12/4) + 2 = -3 + 2 = -1So, the equation is:( (x + 3/2)^2 + (y - √3/2)^2 - 1 = 0 )Therefore, the circle is:( (x + 3/2)^2 + (y - √3/2)^2 = 1 )So, center at (-3/2, √3/2) and radius 1.Now, we need to find the range of ( frac{y}{x} ) for points (x, y) on this circle.This is equivalent to finding the set of all possible values of ( k = y/x ) such that the line y = kx intersects the circle.Alternatively, substituting y = kx into the circle equation and finding for which k the equation has real solutions. The range of k is the set of real numbers for which the equation has at least one real solution (x, y).So, substitute y = kx into the circle equation:( (x + 3/2)^2 + (kx - √3/2)^2 = 1 )Let's expand this:First, expand ( (x + 3/2)^2 ):( x^2 + 3x + 9/4 )Then, expand ( (kx - √3/2)^2 ):( k^2 x^2 - √3 k x + 3/4 )So, adding both expansions:( x^2 + 3x + 9/4 + k^2 x^2 - √3 k x + 3/4 = 1 )Combine like terms:( (1 + k^2) x^2 + (3 - √3 k) x + (9/4 + 3/4 - 1) = 0 )Simplify constants:9/4 + 3/4 = 12/4 = 3, so 3 - 1 = 2. So:( (1 + k^2) x^2 + (3 - √3 k) x + 2 = 0 )This is a quadratic equation in x. For real solutions to exist, the discriminant must be non-negative.Discriminant D:( [3 - √3 k]^2 - 4(1 + k^2)(2) geq 0 )Compute each part:First, expand [3 - √3 k]^2:= 9 - 6√3 k + 3 k^2Then compute 4(1 + k^2)(2):= 8(1 + k^2) = 8 + 8k^2So, discriminant D:9 - 6√3 k + 3k^2 - 8 - 8k^2 ≥ 0Combine like terms:(3k^2 - 8k^2) + (-6√3 k) + (9 - 8) ≥ 0= (-5k^2) - 6√3 k + 1 ≥ 0Multiply both sides by -1 (reversing inequality):5k^2 + 6√3 k - 1 ≤ 0So, solving 5k^2 + 6√3 k - 1 ≤ 0This is a quadratic inequality. Let's find the roots of the quadratic equation 5k^2 + 6√3 k - 1 = 0.Using the quadratic formula:k = [ -6√3 ± √( (6√3)^2 - 4*5*(-1) ) ] / (2*5)Calculate discriminant inside the square root:(6√3)^2 - 4*5*(-1) = 36*3 + 20 = 108 + 20 = 128So, √128 = 8√2Therefore, roots are:k = [ -6√3 ± 8√2 ] / 10Simplify:k = [ -6√3 + 8√2 ] / 10 and k = [ -6√3 - 8√2 ] / 10So, the quadratic expression 5k^2 + 6√3 k - 1 is a parabola opening upwards (since coefficient of k^2 is positive). Therefore, the inequality 5k^2 + 6√3 k - 1 ≤ 0 is satisfied between the two roots.Hence, the range of k is [ ( -6√3 - 8√2 ) / 10 , ( -6√3 + 8√2 ) / 10 ]But since we are asked for the range of ( frac{y}{x} ), which is k, this interval is the answer.But perhaps we can write it in a different form, maybe rationalizing or simplifying.Alternatively, factor out common terms:Let me compute the numerical values to check.First, compute ( -6√3 + 8√2 ) / 10.√3 ≈ 1.732, √2 ≈ 1.414.So:-6*1.732 ≈ -10.3928*1.414 ≈ 11.312So, adding them: -10.392 + 11.312 ≈ 0.92Divide by 10: ≈ 0.092Similarly, the other root:( -6√3 - 8√2 ) / 10 ≈ (-10.392 - 11.312)/10 ≈ (-21.704)/10 ≈ -2.1704So, approximately, the range is from -2.1704 to 0.092. But we need to present the exact form.Alternatively, perhaps factor out a negative sign:Wait, the two roots are:k₁ = [ -6√3 + 8√2 ] / 10k₂ = [ -6√3 - 8√2 ] / 10Alternatively, factor out a 2 from numerator:k₁ = ( -6√3 + 8√2 ) / 10 = ( 2*(-3√3 + 4√2) ) / 10 = (-3√3 + 4√2)/5Similarly, k₂ = ( -6√3 - 8√2 ) / 10 = ( 2*(-3√3 - 4√2) ) /10 = (-3√3 -4√2)/5Therefore, the range of k is [ (-3√3 -4√2)/5 , (-3√3 +4√2)/5 ]But maybe this is the simplest form. Let me check if that's correct.Wait, but I need to confirm my steps once again.Starting from the equation:We had the circle equation, substituted y = kx, obtained a quadratic in x, found discriminant, solved for k, and ended up with the quadratic inequality. So, yes, the roots are correct.But let me check the discriminant calculation again.Original discriminant D:[3 - √3 k]^2 - 4*(1 + k^2)*2 ≥ 0[9 -6√3 k + 3k^2] - 8 -8k^2 ≥0Which becomes:9 -6√3k +3k² -8 -8k² ≥0So, (9 -8) + (-6√3k) + (3k² -8k²) ≥0Which is 1 -6√3k -5k² ≥0Then multiplying by -1:5k² +6√3k -1 ≤0Yes, correct.Then quadratic formula:k = [ -6√3 ± sqrt( (6√3)^2 -4*5*(-1) ) ]/(2*5)Compute discriminant inside sqrt:(6√3)^2 = 36*3=1084*5*(-1) = -20, so -4*5*(-1) = +20Therefore, sqrt(108 +20)=sqrt(128)=8√2Thus, k=( -6√3 ±8√2 ) /10Yes, correct.Hence, the range is [ ( -6√3 -8√2 ) /10 , ( -6√3 +8√2 ) /10 ]But let's see if this can be simplified further. Let me factor numerator:Take numerator as -6√3 ±8√2. Factor out a common factor? Let's see:6 and 8 share a factor of 2. So:= 2*(-3√3 ±4√2)/10 = (-3√3 ±4√2)/5Thus, k ∈ [ (-3√3 -4√2)/5 , (-3√3 +4√2)/5 ]So, written as:Lower bound: (-3√3 -4√2)/5Upper bound: (-3√3 +4√2)/5Alternatively, factor out the negative sign from the lower bound:= - (3√3 +4√2)/5Similarly, upper bound: ( -3√3 +4√2 ) /5But perhaps it's better to leave it as is. So, the range is from [ (-3√3 -4√2)/5 , (-3√3 +4√2)/5 ]Alternatively, since the problem is in Chinese (the original user might be Chinese), sometimes they prefer to rationalize or present it in another form. But in terms of exact form, this seems acceptable.But let me check if there's a mistake here. Because when we substitute y = kx into the circle equation, we might have missed something.Wait, the circle center is (-3/2, √3/2), radius 1.The line from the origin (0,0) to a point (x, y) on the circle. So, the slope is y/x. We need the range of y/x for (x,y) on the circle.Alternatively, parametric equations. Maybe parametrize the circle and then express y/x.But perhaps using geometry: The range of y/x is the set of all slopes from the origin to points on the circle. The maximum and minimum slopes can be found using the tangent lines from the origin to the circle.Yes, that's another approach. Let me try this.Given a circle with center (h, k) = (-3/2, √3/2) and radius r = 1.We need to find the equations of the tangent lines from the origin (0,0) to this circle. The slopes of these tangent lines will give the maximum and minimum values of y/x.The formula for the tangent lines from an external point (x0,y0) to a circle (h,k) with radius r is:The equations can be found by solving the condition that the distance from the point to the line is equal to the radius.Alternatively, the equation of any line through (0,0) is y = kx. The condition that this line is tangent to the circle is that the distance from the center of the circle to the line is equal to the radius.The distance from a point (h, k) to the line ax + by + c =0 is |ah + bk + c| / sqrt(a^2 + b^2). Here, the line is y = kx, which can be rewritten as kx - y =0. So, a =k, b = -1, c=0.Therefore, the distance from center (h, k_c) = (-3/2, √3/2) to the line kx - y =0 is |k*(-3/2) -1*(√3/2) +0| / sqrt(k^2 +1)This distance must be equal to the radius, which is 1.So:| -3k/2 - √3/2 | / sqrt(k^2 +1) =1Multiply both sides by sqrt(k^2 +1):| -3k - √3 | / 2 = sqrt(k^2 +1)Square both sides to eliminate absolute value and square root:( (-3k -√3)^2 ) /4 = k^2 +1Expand numerator:(9k² +6√3 k +3)/4 =k² +1Multiply both sides by 4:9k² +6√3 k +3 =4k² +4Bring all terms to left:9k² +6√3 k +3 -4k² -4=0Simplify:5k² +6√3 k -1=0Which is exactly the same quadratic equation as before! So, this confirms the previous result. Therefore, the solutions are the same.Thus, the range of ( frac{y}{x} ) is [ ( -3√3 -4√2 ) /5 , ( -3√3 +4√2 ) /5 ]Hence, the answer should be written in this interval.But let me check the arithmetic again, just to be sure.Original equation after squaring:9k² +6√3 k +3 =4k² +4Subtract 4k² and 4:5k² +6√3 k -1=0Yes, that's correct. Then quadratic formula:k = [ -6√3 ± sqrt( (6√3)^2 -4*5*(-1) ) ]/(2*5)Which is [ -6√3 ± sqrt(108 +20) ]/10 = [ -6√3 ± sqrt(128) ]/10 = [ -6√3 ± 8√2 ]/10So, yes, same as before. So, factoring numerator gives:= [ -6√3 ±8√2 ] /10 = [ -3√3 ±4√2 ] /5Therefore, the range is [ (-3√3 -4√2)/5 , (-3√3 +4√2)/5 ]So, this is the exact form. If needed, we can write this as an interval:( left[ dfrac{ -3sqrt{3} -4sqrt{2} }{5}, dfrac{ -3sqrt{3} +4sqrt{2} }{5} right] )But let me check the signs again. The lower bound is when we take the negative sign, so:(-3√3 -4√2)/5 is the lower bound, and (-3√3 +4√2)/5 is the upper bound. Since 4√2 is positive and larger in magnitude than 3√3?Wait, 3√3 ≈ 3*1.732≈5.196, 4√2≈4*1.414≈5.656. So, 4√2 is larger than 3√3, so (-3√3 +4√2)/5 ≈ (-5.196 +5.656)/5≈0.46/5≈0.092, which is positive, and (-3√3 -4√2)/5≈(-5.196 -5.656)/5≈-10.852/5≈-2.170, which is negative. So the range is from approximately -2.17 to 0.09, which makes sense because the circle is located in the left half-plane (center at x=-3/2), so the lines from the origin to points on the circle can have both negative and positive slopes, but not too steep.Wait, but the upper bound is positive, which is 0.09, and lower bound is -2.17, so this seems correct.But just to confirm with another approach: Let's parametrize the circle.Parametrize the circle as:x = -3/2 + cosθy = √3/2 + sinθSince the center is (-3/2, √3/2) and radius 1.Then, we need to find the range of (y)/(x) = [√3/2 + sinθ]/[-3/2 + cosθ]So, k = [√3/2 + sinθ]/[-3/2 + cosθ]We can consider this as a function of θ and find its maximum and minimum.Alternatively, this seems more complicated, but maybe we can set t = tan(θ/2) and use substitution. But perhaps calculus.Let me set θ as a parameter and find the derivative of k with respect to θ and set to zero. But this might be tedious. However, let's try.Let k(θ) = [√3/2 + sinθ]/[-3/2 + cosθ]To find extrema, take derivative dk/dθ and set to zero.First, let me compute dk/dθ:Let numerator = N = √3/2 + sinθDenominator = D = -3/2 + cosθThen, dk/dθ = (D * cosθ - N * (-sinθ)) / D²= [ D cosθ + N sinθ ] / D²Set derivative to zero:D cosθ + N sinθ =0Substitute N and D:(-3/2 + cosθ) cosθ + (√3/2 + sinθ) sinθ =0Expand:-3/2 cosθ + cos²θ + √3/2 sinθ + sin²θ =0Combine cos²θ + sin²θ =1:-3/2 cosθ +1 + √3/2 sinθ =0So,-3/2 cosθ + √3/2 sinθ = -1Multiply both sides by 2:-3 cosθ + √3 sinθ = -2Let me write this as:√3 sinθ -3 cosθ = -2This is of the form A sinθ + B cosθ = CWhich can be rewritten as R sin(θ + φ) = C, where R = sqrt(A² + B²), tanφ = B/A.Here, A = √3, B = -3Thus, R = sqrt( (√3)^2 + (-3)^2 ) = sqrt(3 +9) = sqrt(12) = 2*sqrt(3)φ = arctan(B/A) = arctan(-3/√3) = arctan(-√3) = -π/3Hence, the equation becomes:2√3 sin(θ - π/3) = -2Divide both sides by 2√3:sin(θ - π/3) = -1/√3Thus,θ - π/3 = arcsin(-1/√3) + 2πn or π - arcsin(-1/√3) + 2πnSimplify:θ = π/3 + arcsin(-1/√3) + 2πnor θ = π/3 + π - arcsin(-1/√3) + 2πnBut arcsin(-1/√3) = -arcsin(1/√3)Therefore,θ = π/3 - arcsin(1/√3) + 2πnor θ = π/3 + π + arcsin(1/√3) + 2πnSimplify further:θ = π/3 - α + 2πn or θ = 4π/3 + α + 2πn, where α = arcsin(1/√3)Compute α:arcsin(1/√3) is an angle whose sine is 1/√3 ≈0.577, so α ≈0.6155 radians ≈35.264 degreesTherefore, the critical points are at θ ≈ π/3 -0.6155 ≈1.047 -0.6155≈0.431 radians and θ≈4π/3 +0.6155≈4.188 +0.6155≈4.803 radians.Now, we need to compute k(θ) at these critical angles to find the maximum and minimum.First, compute θ ≈0.431 radians:Compute N = √3/2 + sinθ ≈0.866 + sin(0.431)≈0.866 +0.418≈1.284D = -3/2 + cosθ≈-1.5 + cos(0.431)≈-1.5 +0.908≈-0.592So, k≈1.284 / -0.592≈-2.169Second, compute θ≈4.803 radians:N = √3/2 + sinθ≈0.866 + sin(4.803)≈0.866 + sin(4.803)But 4.803 radians is more than π (3.1416), so in the fourth quadrant. sin(4.803)≈sin(4.803 - 2π)≈sin(4.803 -6.283)≈sin(-1.480)≈-sin(1.480)≈-0.996Wait, let's compute sin(4.803):4.803 radians is 4.803 - π ≈4.803 -3.142≈1.661 radians, which is in the third quadrant. Wait, no:Wait, 4.803 radians is more than π (3.1416) but less than 2π (6.283). Specifically, 4.803 - π ≈1.661 radians, which is π/2 + ~0.57 radians, so in the third quadrant.But sin(4.803) = sin(π +1.661) = -sin(1.661)≈-0.996Therefore, N≈0.866 -0.996≈-0.130D = -1.5 + cos(4.803). cos(4.803)=cos(π +1.661)= -cos(1.661)≈-cos(1.661). cos(1.661)≈-0.095Therefore, cos(4.803)≈-(-0.095)=0.095So, D≈-1.5 +0.095≈-1.405Thus, k≈-0.130 / -1.405≈0.092Which matches our earlier approximate values. Hence, the maximum value of k is approximately 0.092 and the minimum is approximately -2.169, which correspond to the exact values ( -3√3 +4√2 ) /5 and ( -3√3 -4√2 ) /5.Therefore, the exact range is [ ( -3√3 -4√2 ) /5 , ( -3√3 +4√2 ) /5 ]So, the answer should be written in boxed notation as boxed{left[ dfrac{-3sqrt{3} - 4sqrt{2}}{5}, dfrac{-3sqrt{3} + 4sqrt{2}}{5} right]}Final AnswerThe range of (frac{y}{x}) is boxed{left[ dfrac{-3sqrt{3} - 4sqrt{2}}{5}, dfrac{-3sqrt{3} + 4sqrt{2}}{5} right]}.</think>Given the complex number ( z = x + yi ) satisfies the condition that the ratio of the real part to the imaginary part of ( frac{z + 1}{z + 2} ) is ( sqrt{3} ), we need to find the range of ( frac{y}{x} ).First, we express ( frac{z + 1}{z + 2} ) in terms of its real and imaginary parts. Let ( z = x + yi ), then:[frac{z + 1}{z + 2} = frac{(x + 1) + yi}{(x + 2) + yi}]Multiplying the numerator and denominator by the conjugate of the denominator, we get:[frac{(x + 1) + yi}{(x + 2) + yi} cdot frac{(x + 2) - yi}{(x + 2) - yi} = frac{(x + 1)(x + 2) + y^2 + yi}{(x + 2)^2 + y^2}]Separating the real and imaginary parts, we have:[text{Real part} = frac{x^2 + 3x + 2 + y^2}{(x + 2)^2 + y^2}, quad text{Imaginary part} = frac{y}{(x + 2)^2 + y^2}]Given the ratio of the real part to the imaginary part is ( sqrt{3} ):[frac{frac{x^2 + 3x + 2 + y^2}{(x + 2)^2 + y^2}}{frac{y}{(x + 2)^2 + y^2}} = sqrt{3} implies frac{x^2 + 3x + 2 + y^2}{y} = sqrt{3}]Rearranging, we get:[x^2 + y^2 + 3x + 2 - sqrt{3}y = 0]This is the equation of a circle. Converting it to standard form by completing the square:[left( x + frac{3}{2} right)^2 + left( y - frac{sqrt{3}}{2} right)^2 = 1]The circle has center ( left( -frac{3}{2}, frac{sqrt{3}}{2} right) ) and radius 1. To find the range of ( frac{y}{x} ), we consider the slopes of lines from the origin to points on this circle. Using the quadratic equation derived from substituting ( y = kx ) into the circle equation and ensuring the discriminant is non-negative, we solve:[5k^2 + 6sqrt{3}k - 1 leq 0]Solving this quadratic inequality using the quadratic formula, we find the roots:[k = frac{-6sqrt{3} pm 8sqrt{2}}{10} = frac{-3sqrt{3} pm 4sqrt{2}}{5}]Thus, the range of ( frac{y}{x} ) is:[boxed{left[ dfrac{-3sqrt{3} - 4sqrt{2}}{5}, dfrac{-3sqrt{3} + 4sqrt{2}}{5} right]}]]

🔑:4. left[frac{-3 sqrt{3}-4 sqrt{2}}{5}, frac{-3 sqrt{3}+4 sqrt{2}}{5}right]Analysis: frac{z+1}{z+2}=frac{x+1+y i}{x+2+y i}=frac{left(x^{2}+3 x+2+y^{2}right)+y i}{(x+2)^{2}+y^{2}}, from the problem, we get x^{2}+3 x+2+y^{2}=sqrt{3} y which is: left(x+frac{3}{2}right)^{2}+left(y-frac{sqrt{3}}{2}right)^{2}=1, let y=k x, then left(k^{2}+1right) x^{2}+(3-sqrt{3} k) x+2=0. By Delta geq 0, we have 9+3 k^{2}-6 sqrt{3} k geq 8 k^{2}+8 Leftrightarrow y inleft[frac{-3 sqrt{3}-4 sqrt{2}}{5}, frac{-3 sqrt{3}+4 sqrt{2}}{5}right].

🧠:[<think>Okay, let's see. I need to solve these two parts. First, for part (1), if the line x=a is an axis of symmetry for the graph of y = f(x) = cos²x, then I need to find g(2a). Hmm. Alright, let's start by understanding what it means for x=a to be an axis of symmetry for a function's graph. Symmetry about the line x=a means that for any point (a + h, y) on the graph, there is a corresponding point (a - h, y). So, mathematically, this translates to f(a + h) = f(a - h) for all h. That's the definition of a vertical line of symmetry at x=a. So, applying this to f(x) = cos²x, which is the function given.So, the function f(x) must satisfy f(a + h) = f(a - h) for all h. Let's write that out. f(a + h) = cos²(a + h) and f(a - h) = cos²(a - h). So, cos²(a + h) must equal cos²(a - h) for all h. Hmm. When does that happen? Well, cos² is an even function, so maybe if the argument inside the cosine has some symmetry. Let me recall that cos(a + h) = cos a cos h - sin a sin h, and cos(a - h) = cos a cos h + sin a sin h. So, their squares would be [cos a cos h - sin a sin h]^2 and [cos a cos h + sin a sin h]^2. Let's compute these:First, [cos a cos h - sin a sin h]^2 = cos²a cos²h - 2 cos a cos h sin a sin h + sin²a sin²hSimilarly, [cos a cos h + sin a sin h]^2 = cos²a cos²h + 2 cos a cos h sin a sin h + sin²a sin²hSo, the difference between these two is the middle term: -4 cos a cos h sin a sin h. For these squares to be equal for all h, this difference must be zero. So, we have:[cos²(a + h) - cos²(a - h)] = -4 cos a sin a cos h sin h = -2 sin(2a) cos h sin h = 0 for all h.Wait, because cos²(a + h) = cos²(a - h) for all h implies that their difference is zero for all h. Therefore, -2 sin(2a) cos h sin h = 0 for all h. But cos h sin h is not zero for all h; for example, when h = π/4, cos h sin h = √2/2 * √2/2 = 1/2 ≠ 0. So, the only way this can hold for all h is if sin(2a) = 0. Therefore, sin(2a) = 0. Solving this equation, we get 2a = nπ, where n is an integer. Therefore, a = nπ/2. So, the possible values of a are multiples of π/2. But wait, since a is an axis of symmetry, maybe only certain values of a are possible here. Let me check.But wait, if a = 0, then the axis of symmetry is x=0. Is the graph of y = cos²x symmetric about x=0? Yes, because cos²x is an even function. Similarly, if a = π/2, then x=π/2 is an axis of symmetry. Let's check if cos²x is symmetric about x=π/2. So, for x=π/2 + h and x=π/2 - h, we need cos²(π/2 + h) = cos²(π/2 - h). But cos(π/2 + h) = -sin h and cos(π/2 - h) = sin h. Therefore, cos²(π/2 + h) = sin²h and cos²(π/2 - h) = sin²h. So, they are equal. Hence, x=π/2 is also an axis of symmetry. Similarly, a = π, then x=π is an axis of symmetry. But cos²(π + h) = cos²(π - h). Let's check: cos(π + h) = -cos h and cos(π - h) = -cos h, so their squares are equal. So, x=π is also an axis of symmetry. Therefore, all a = nπ/2, n integer, are axes of symmetry. Therefore, the possible values of a are multiples of π/2.But the problem says "the line x=a is an axis of symmetry". So, in general, the axes of symmetry for cos²x are all x = nπ/2, n integer. Therefore, a = nπ/2. So, then 2a = nπ. Therefore, g(2a) = g(nπ). Let's compute g(nπ). Given that g(x) = 1/2 + √3 sin x cos x. Let's compute sin x cos x at x = nπ. Since sin(nπ) = 0 for all integers n, so sin x cos x = 0. Therefore, g(nπ) = 1/2 + √3 * 0 = 1/2. Therefore, regardless of n, g(2a) = 1/2. Therefore, the value is 1/2. So, the answer for part (1) is 1/2.Wait, but is that correct? Let me double-check. Let's take a specific example. Let a = 0, which is an axis of symmetry. Then 2a = 0, and g(0) = 1/2 + √3 sin 0 cos 0 = 1/2 + 0 = 1/2. Correct. If a = π/2, then 2a = π. g(π) = 1/2 + √3 sin π cos π = 1/2 + 0 = 1/2. Similarly, a = π, 2a = 2π, g(2π) = 1/2 + √3 sin 2π cos 2π = 1/2 + 0 = 1/2. So, yes, it seems correct. Therefore, the answer to part (1) is 1/2.Moving on to part (2): If 0 ≤ x ≤ π/2, find the range of h(x) = f(x) + g(x). So, h(x) = cos²x + 1/2 + √3 sinx cosx. Let's write that out: h(x) = cos²x + 1/2 + √3 sinx cosx. Let's see if we can simplify this expression.First, maybe express cos²x in terms of double angle. Since cos²x = (1 + cos 2x)/2. So, substituting that in, we get h(x) = (1 + cos 2x)/2 + 1/2 + √3 sinx cosx. Then, combine the constants: (1/2 + 1/2) = 1. So, h(x) = 1 + (cos 2x)/2 + √3 sinx cosx. But sinx cosx is (sin 2x)/2. Therefore, h(x) = 1 + (cos 2x)/2 + √3 (sin 2x)/2. So, h(x) = 1 + (1/2) cos 2x + (√3/2) sin 2x. Now, this looks like 1 plus a combination of sin and cos terms. Let's factor that combination as a single sine or cosine function. Let me recall that A cos θ + B sin θ = C cos(θ - φ), where C = √(A² + B²) and tan φ = B/A.In this case, A = 1/2 and B = √3/2. So, C = √( (1/2)^2 + (√3/2)^2 ) = √(1/4 + 3/4) = √1 = 1. Then, tan φ = (√3/2)/(1/2) = √3. Therefore, φ = π/3. Therefore, h(x) = 1 + cos(2x - π/3). Wait, let me check:A cos θ + B sin θ = C cos(θ - φ). So here, θ is 2x. So, (1/2) cos 2x + (√3/2) sin 2x = cos(2x - π/3). Because cos π/3 = 1/2 and sin π/3 = √3/2. Therefore, cos(2x - π/3) = cos 2x cos π/3 + sin 2x sin π/3 = (1/2) cos 2x + (√3/2) sin 2x. Exactly. Therefore, h(x) = 1 + cos(2x - π/3).Therefore, h(x) can be written as 1 + cos(2x - π/3). Now, to find the range of h(x) over the interval [0, π/2], we can analyze this expression. The cosine function varies between -1 and 1, so 1 + cos(...) varies between 0 and 2. However, the argument inside the cosine is 2x - π/3, so we need to consider how 2x - π/3 varies as x goes from 0 to π/2. Let's compute the interval for 2x - π/3.When x = 0: 2(0) - π/3 = -π/3When x = π/2: 2(π/2) - π/3 = π - π/3 = 2π/3Therefore, the argument 2x - π/3 ranges from -π/3 to 2π/3. Therefore, we need to find the maximum and minimum of cos(θ) where θ ∈ [-π/3, 2π/3]. Let's recall that the cosine function is decreasing from 0 to π and increasing from π to 2π, but here our interval is from -π/3 to 2π/3. Let's sketch the cosine graph over this interval.From θ = -π/3 (which is equivalent to 5π/3) to θ = 0, cosine increases from cos(-π/3) = cos(π/3) = 1/2 to cos(0) = 1. Then, from θ = 0 to θ = π, cosine decreases from 1 to -1. However, our upper limit is θ = 2π/3, which is less than π. So, from θ = 0 to θ = 2π/3, cosine decreases from 1 to cos(2π/3) = -1/2. Therefore, over the interval [-π/3, 2π/3], the maximum value of cosθ is 1 (at θ=0), and the minimum is -1/2 (at θ=2π/3). However, wait, when θ = -π/3, cosθ = 1/2. Then, as θ increases from -π/3 to 0, cosθ increases to 1. Then, as θ increases from 0 to 2π/3, cosθ decreases to -1/2. Therefore, the maximum value of cosθ is 1, and the minimum is -1/2. Therefore, the range of cosθ over θ ∈ [-π/3, 2π/3] is [-1/2, 1]. Therefore, h(x) = 1 + cosθ ranges from 1 + (-1/2) = 1/2 to 1 + 1 = 2.But wait, is that correct? Let me confirm. If θ ranges from -π/3 to 2π/3, then:At θ = -π/3: cosθ = 1/2, so h(x) = 1 + 1/2 = 3/2At θ = 0: cosθ = 1, so h(x) = 2At θ = 2π/3: cosθ = -1/2, so h(x) = 1 - 1/2 = 1/2But between θ = -π/3 and θ = 0, cosθ is increasing from 1/2 to 1, so h(x) increases from 3/2 to 2. Then, from θ = 0 to θ = 2π/3, cosθ decreases from 1 to -1/2, so h(x) decreases from 2 to 1/2. Therefore, over the entire interval θ ∈ [-π/3, 2π/3], h(x) attains maximum 2 at θ = 0 (which corresponds to x = π/6, since 2x - π/3 = 0 ⇒ x = π/6) and minimum 1/2 at θ = 2π/3 (which corresponds to x = π/2). However, at θ = -π/3 (x=0), h(x) = 3/2. So, the range is from 1/2 to 2. But wait, when x=0, h(x) is 3/2, and when x=π/6, it's 2, and when x=π/2, it's 1/2. Therefore, the maximum value is 2, the minimum is 1/2. Therefore, the range of h(x) is [1/2, 2]. But wait, but at x=0, h(x) = cos²0 + 1/2 + √3 sin0 cos0 = 1 + 1/2 + 0 = 3/2. Similarly, let's compute h(0): f(0) + g(0) = 1 + 1/2 + 0 = 3/2. Similarly, h(π/6) = cos²(π/6) + 1/2 + √3 sin(π/6) cos(π/6). Compute cos(π/6) = √3/2, so cos²(π/6) = 3/4. sin(π/6) = 1/2, cos(π/6) = √3/2, so sin(π/6) cos(π/6) = (1/2)(√3/2) = √3/4. Multiply by √3: √3*(√3/4) = 3/4. Therefore, h(π/6) = 3/4 + 1/2 + 3/4 = 3/4 + 3/4 = 3/2 + 1/2 = 2. Yes, that's correct. At x=π/6, h(x) is 2.At x=π/2: f(π/2) = cos²(π/2) = 0, g(π/2) = 1/2 + √3 sin(π/2) cos(π/2) = 1/2 + √3*1*0 = 1/2. So, h(π/2) = 0 + 1/2 = 1/2. Therefore, the minimum is 1/2. However, at x=0, h(x) is 3/2, which is between 1/2 and 2. Therefore, the function h(x) starts at 3/2 when x=0, goes up to 2 at x=π/6, then decreases to 1/2 at x=π/2. Therefore, the range is [1/2, 2]. Wait, but according to the previous analysis, the maximum is 2 and the minimum is 1/2, so the range is [1/2, 2]. But when x=0, h(x) = 3/2, which is inside that interval. Therefore, the full range is from the minimum 1/2 to maximum 2. Therefore, the answer is [1/2, 2]. But let me confirm once more.Alternatively, since h(x) = 1 + cos(2x - π/3), and 2x - π/3 ranges from -π/3 to 2π/3, as we saw. The maximum of cos(θ) in this interval is 1, and the minimum is -1/2. Therefore, h(x) ranges from 1 + (-1/2) = 1/2 to 1 + 1 = 2. So, yes, that's correct. Therefore, the range is [1/2, 2].But just to make sure, let's compute h(x) at another point. Let's take x=π/3. Then 2x - π/3 = 2π/3 - π/3 = π/3. So, cos(π/3) = 1/2. Therefore, h(π/3) = 1 + 1/2 = 3/2. So, same as at x=0, h(x) is 3/2. So, between x=0 and x=π/6, h(x) increases from 3/2 to 2, then decreases back to 3/2 at x=π/3, then continues decreasing to 1/2 at x=π/2. Wait, hold on. Wait, at x=π/3, which is 60 degrees, h(x) is 3/2? Let's compute directly:f(π/3) = cos²(π/3) = (1/2)^2 = 1/4g(π/3) = 1/2 + √3 sin(π/3) cos(π/3) = 1/2 + √3*(√3/2)*(1/2) = 1/2 + √3*(√3/4) = 1/2 + 3/4 = 5/4Therefore, h(π/3) = 1/4 + 5/4 = 6/4 = 3/2. Correct. So, that's consistent. Therefore, the function h(x) starts at 3/2 at x=0, peaks at 2 when x=π/6, then goes back down to 3/2 at x=π/3, then continues decreasing to 1/2 at x=π/2. So, the maximum is 2, and the minimum is 1/2. Therefore, the range is [1/2, 2]. So, that's the answer for part (2).Therefore, summarizing:1) For part (1), the value of g(2a) is 1/2.2) For part (2), the range of h(x) is [1/2, 2].Final Answer(1) boxed{dfrac{1}{2}} (2) boxed{left[ dfrac{1}{2}, 2 right]}</think>Given functions ( f(x) = cos^2 x ) and ( g(x) = frac{1}{2} + sqrt{3} sin x cos x ):(1) If the line ( x = a ) is an axis of symmetry for the graph of ( y = f(x) ), we need to find the value of ( g(2a) ). For ( x = a ) to be an axis of symmetry, ( f(a + h) = f(a - h) ) for all ( h ). This implies ( cos^2(a + h) = cos^2(a - h) ) for all ( h ). This is true if ( sin(2a) = 0 ), leading to ( 2a = npi ) for integer ( n ). Thus, ( a = frac{npi}{2} ). Therefore, ( 2a = npi ), and ( g(npi) = frac{1}{2} + sqrt{3} sin(npi) cos(npi) = frac{1}{2} ). Hence, the value of ( g(2a) ) is (boxed{frac{1}{2}}).(2) To find the range of ( h(x) = f(x) + g(x) ) for ( 0 leq x leq frac{pi}{2} ), we start by expressing ( h(x) ) in a simplified form:[h(x) = cos^2 x + frac{1}{2} + sqrt{3} sin x cos x]Using trigonometric identities, we rewrite ( h(x) ):[h(x) = 1 + cos(2x - frac{pi}{3})]The argument ( 2x - frac{pi}{3} ) ranges from ( -frac{pi}{3} ) to ( frac{2pi}{3} ) as ( x ) varies from ( 0 ) to ( frac{pi}{2} ). The cosine function over this interval ranges from (-1/2) to (1). Therefore, ( h(x) ) ranges from ( 1 - frac{1}{2} = frac{1}{2} ) to ( 1 + 1 = 2 ). Hence, the range of ( h(x) ) is (boxed{left[ frac{1}{2}, 2 right]}).,<think>Okay, let's try to tackle these two problems step by step. First, problem (1): If the line x=a is an axis of symmetry for the graph of y=f(x), find the value of g(2a). Hmm, okay. So f(x) is cos²x, and g(x) is 1/2 plus sqrt(3) sinx cosx. So, first, I need to figure out what value of 'a' makes x=a an axis of symmetry for y = cos²x. Then, once I find that 'a', plug 2a into g(x) to find g(2a). Alright. Symmetry about the line x=a means that for any point (a + h, y) on the graph, there's a corresponding point (a - h, y). So, mathematically, this would mean that f(a + h) = f(a - h) for all h. So, the function is symmetric around x=a. Given that f(x) is cos²x, let's write down the equation f(a + h) = f(a - h). So, cos²(a + h) = cos²(a - h). But cos² is an even function, right? Because cos(-θ) = cosθ, so cos²(-θ) = cos²θ. So, maybe this identity can help. Let's see. Expanding cos(a + h) and cos(a - h). Using the formula cos(A ± B) = cosA cosB ∓ sinA sinB. So, cos(a + h) = cos a cos h - sin a sin h, and cos(a - h) = cos a cos h + sin a sin h. Therefore, cos²(a + h) = [cos a cos h - sin a sin h]^2 = cos²a cos²h - 2 cos a cos h sin a sin h + sin²a sin²h. Similarly, cos²(a - h) = [cos a cos h + sin a sin h]^2 = cos²a cos²h + 2 cos a cos h sin a sin h + sin²a sin²h. Setting these equal: cos²a cos²h - 2 cos a sin a cos h sin h + sin²a sin²h = cos²a cos²h + 2 cos a sin a cos h sin h + sin²a sin²h. Subtracting the left side from the right side, we get 4 cos a sin a cos h sin h = 0. So, 4 cos a sin a cos h sin h = 0 for all h. But this must hold for all h. The term 4 cos a sin a cos h sin h must be zero for all h. But cos h sin h is not identically zero; for example, when h = π/4, cos h sin h = (sqrt(2)/2)(sqrt(2)/2) = 1/2 ≠ 0. Therefore, the only way for 4 cos a sin a cos h sin h to be zero for all h is if cos a sin a = 0. So, cos a sin a = 0. This implies either cos a = 0 or sin a = 0. Solving cos a = 0: a = π/2 + kπ for integer k. Solving sin a = 0: a = kπ for integer k. Therefore, the lines x=a that are axes of symmetry for y = cos²x are the vertical lines at multiples of π/2. Wait, but cos²x has a period of π, right? Because cos²x = (1 + cos2x)/2, so the period is π. So, the graph of cos²x is a cosine wave with period π, oscillating between 0 and 1. Therefore, the function y = cos²x should have axes of symmetry at each peak and trough. Since the peaks are at x = 0, π, 2π, etc., and the troughs are at x = π/2, 3π/2, etc. So, indeed, the axes of symmetry are at x = kπ/2, where k is an integer. Wait, but if we think about the standard cosine squared graph, it's symmetric about every vertical line through its maxima and minima. So, maxima at x=0, π, 2π,... minima at x=π/2, 3π/2, 5π/2,... So, yes, the axes of symmetry would be at those points. Therefore, the values of a are integer multiples of π/2. But wait, the problem says "the line x=a is an axis of symmetry". So, a can be any multiple of π/2. But the problem doesn't specify a particular interval for a, so in general, a = kπ/2 where k is integer. But since we need to compute g(2a), let's see. Let's take a general a = kπ/2, then 2a = kπ. Then, g(2a) = 1/2 + sqrt(3) sin(2a) cos(2a). Wait, but 2a = kπ, so sin(kπ) = 0 for any integer k. Therefore, sin(kπ) cos(kπ) = 0 * (-1)^k = 0. Therefore, g(2a) = 1/2 + 0 = 1/2. Wait, that's interesting. So regardless of k, g(2a) = 1/2. Therefore, the answer to part (1) is 1/2. But let me verify this again. Let's take a specific example. If a = 0, which is a point of symmetry (the maximum at x=0). Then 2a = 0, so g(0) = 1/2 + sqrt(3) sin0 cos0 = 1/2 + 0 = 1/2. If a = π/2, which is a trough, so x=π/2 is also an axis of symmetry. Then 2a = π. So g(π) = 1/2 + sqrt(3) sinπ cospi = 1/2 + 0 = 1/2. Similarly, a = π, 2a = 2π, g(2π) = 1/2 + sqrt(3) sin2π cos2π = 1/2. So yes, regardless of the axis of symmetry, g(2a) is always 1/2. So the answer is 1/2. Alright, that seems straightforward once we recognize that the axes of symmetry for cos²x are at multiples of π/2, leading to 2a being multiples of π, and thus the sine term in g(2a) becomes zero. Now, moving on to problem (2): If 0 ≤ x ≤ π/2, find the range of h(x) = f(x) + g(x). So h(x) = cos²x + 1/2 + sqrt(3) sinx cosx. Let's simplify this expression. First, let's write down h(x):h(x) = cos²x + 1/2 + sqrt(3) sinx cosx.We can use the identity cos²x = (1 + cos2x)/2. Let's substitute that in:h(x) = (1 + cos2x)/2 + 1/2 + sqrt(3) sinx cosx.Simplify the constants: (1/2 + 1/2) = 1. So,h(x) = 1 + (cos2x)/2 + sqrt(3) sinx cosx.Also, note that sinx cosx = (sin2x)/2. Therefore, sqrt(3) sinx cosx = sqrt(3)/2 sin2x. So substituting that in:h(x) = 1 + (cos2x)/2 + (sqrt(3)/2) sin2x.So h(x) = 1 + (1/2) cos2x + (sqrt(3)/2) sin2x.Hmm, this expression resembles a linear combination of sin2x and cos2x. Let's write it as:h(x) = 1 + (1/2) cos2x + (sqrt(3)/2) sin2x.Recall that A cosθ + B sinθ can be written as C cos(θ - φ), where C = sqrt(A² + B²) and tanφ = B/A. Let's apply that here. Let's set θ = 2x, A = 1/2, B = sqrt(3)/2.Then, C = sqrt( (1/2)^2 + (sqrt(3)/2)^2 ) = sqrt(1/4 + 3/4) = sqrt(1) = 1.And tanφ = B/A = (sqrt(3)/2)/(1/2) = sqrt(3). Therefore, φ = π/3. Because tan(π/3) = sqrt(3).Therefore, h(x) can be rewritten as:h(x) = 1 + cos(2x - π/3).Because A cosθ + B sinθ = cos(θ - φ). So, substituting back:(1/2) cos2x + (sqrt(3)/2) sin2x = cos(2x - π/3). Therefore,h(x) = 1 + cos(2x - π/3).So now, h(x) is 1 plus a cosine function. The range of h(x) will depend on the range of cos(2x - π/3) over the interval x ∈ [0, π/2]. Let's analyze that.First, let's determine the interval for 2x - π/3 when x is in [0, π/2]. When x = 0: 2*0 - π/3 = -π/3.When x = π/2: 2*(π/2) - π/3 = π - π/3 = 2π/3.Therefore, the argument of the cosine, 2x - π/3, ranges from -π/3 to 2π/3 as x goes from 0 to π/2.We need to find the range of cos(θ) where θ ∈ [-π/3, 2π/3].The cosine function is maximum at θ = 0 (cos0 = 1) and decreases as θ moves away from 0 in both directions. Let's check the critical points:At θ = -π/3: cos(-π/3) = cos(π/3) = 1/2.At θ = 0: cos0 = 1.At θ = 2π/3: cos(2π/3) = -1/2.So, the maximum value of cosθ in this interval is 1, and the minimum is -1/2. Therefore, the range of cosθ is [-1/2, 1].Therefore, the range of h(x) = 1 + cosθ is [1 - 1/2, 1 + 1] = [1/2, 2].Wait, that seems straightforward, but let me verify by checking the endpoints and critical points.Alternatively, since h(x) = 1 + cos(2x - π/3), we can find its maximum and minimum over x ∈ [0, π/2].The maximum of cos(2x - π/3) is 1, achieved when 2x - π/3 = 2kπ. Solving for x: 2x = π/3 + 2kπ => x = π/6 + kπ. But since x ∈ [0, π/2], k=0 gives x=π/6. So at x=π/6, cos(2*(π/6) - π/3) = cos(π/3 - π/3) = cos0 = 1. So h(π/6) = 1 + 1 = 2.The minimum of cos(2x - π/3) occurs at the endpoints or where derivative is zero. Let's check when derivative is zero. Let's compute derivative of h(x):h'(x) = -sin(2x - π/3)*2.Setting h'(x) = 0: -2 sin(2x - π/3) = 0 => sin(2x - π/3) = 0.Solutions: 2x - π/3 = kπ => x = (kπ + π/3)/2. Within x ∈ [0, π/2], check possible k:k=0: x = (0 + π/3)/2 = π/6 ≈ 0.523, which is within [0, π/2].k=1: x = (π + π/3)/2 = (4π/3)/2 = 2π/3 ≈ 2.094, which is outside of [0, π/2].So the only critical point is x=π/6, which we already considered as the maximum. Therefore, the minimum occurs at the endpoints x=0 or x=π/2.Compute h(0): h(0) = 1 + cos(0 - π/3) = 1 + cos(-π/3) = 1 + cos(π/3) = 1 + 1/2 = 3/2.Compute h(π/2): h(π/2) = 1 + cos(2*(π/2) - π/3) = 1 + cos(π - π/3) = 1 + cos(2π/3) = 1 + (-1/2) = 1/2.Wait, so at x=π/2, h(x) = 1/2. But earlier, when we considered the range of cosθ over θ ∈ [-π/3, 2π/3], the minimum was -1/2, so h(x) = 1 + (-1/2) = 1/2, which is correct. But h(0) was 3/2, which is 1 + 1/2, but according to the previous analysis, the maximum of cosθ is 1, leading to h(x)=2, which is achieved at x=π/6. So the function reaches a maximum of 2 at x=π/6, a value of 3/2 at x=0, and a minimum of 1/2 at x=π/2. Therefore, the range is from 1/2 to 2. Wait, but this seems conflicting. Wait, when θ ranges from -π/3 to 2π/3, cosθ attains maximum 1 at θ=0, and minimum -1/2 at θ=2π/3. Therefore, 1 + cosθ would range from 1 - 1/2 = 1/2 to 1 + 1 = 2. So the overall range is [1/2, 2]. However, when we compute h(0), we get 3/2, which is within [1/2, 2]. Similarly, h(π/6)=2, h(π/2)=1/2, and h(0)=3/2. So the function attains all values between 1/2 and 2. Therefore, the range is [1/2, 2]. But wait, hold on. The function h(x) is 1 + cos(2x - π/3). The argument 2x - π/3 goes from -π/3 to 2π/3 as x goes from 0 to π/2. The cosine function over this interval starts at cos(-π/3) = 1/2, goes up to cos(0) = 1, then down to cos(2π/3) = -1/2. Therefore, the maximum value of cos is 1, and the minimum is -1/2. Hence, h(x) ranges from 1 + (-1/2) = 1/2 up to 1 + 1 = 2. So the range is [1/2, 2]. But when x=0, h(0) = 1 + cos(-π/3) = 1 + 1/2 = 3/2. So 3/2 is in the middle of the range. Similarly, at x=π/3, for example, which is within [0, π/2], let's check:x=π/3: h(π/3) = 1 + cos(2*(π/3) - π/3) = 1 + cos(π/3) = 1 + 1/2 = 3/2. So yeah, same as at x=0. But the maximum is at x=π/6, which gives h(π/6)=2, and the minimum is at x=π/2, which is 1/2. So the function h(x) indeed ranges from 1/2 to 2. Therefore, the range is [1/2, 2]. Alternatively, another approach is to express h(x) as a single sinusoidal function and then find its amplitude and phase shift, but we've already done that. Alternatively, we can compute derivative and find maxima and minima. Let's do that quickly to confirm. h(x) = 1 + cos(2x - π/3)h'(x) = -2 sin(2x - π/3)Setting h'(x) = 0:-2 sin(2x - π/3) = 0sin(2x - π/3) = 0Solutions: 2x - π/3 = kπ => x = (kπ + π/3)/2Within x ∈ [0, π/2], k=0 gives x=(π/3)/2=π/6 ≈0.523, which is within [0, π/2].k=1 gives x=(π + π/3)/2=(4π/3)/2=2π/3≈2.094, which is outside.So only critical point is x=π/6.At x=π/6:h(π/6) = 1 + cos(2*(π/6) - π/3)=1 + cos(π/3 - π/3)=1 + cos0=1 +1=2.Then check endpoints:x=0: h(0)=1 + cos(-π/3)=1 + 1/2=3/2x=π/2: h(π/2)=1 + cos(2*(π/2)-π/3)=1 + cos(π - π/3)=1 + cos(2π/3)=1 -1/2=1/2Thus, the maximum is 2, the minimum is 1/2. Therefore, the range is [1/2, 2].Therefore, the answer to part (2) is [1/2, 2].So summarizing:1) The value of g(2a) is 1/2.2) The range of h(x) is [1/2, 2].Final Answer(1) boxed{dfrac{1}{2}} (2) boxed{left[ dfrac{1}{2}, 2 right]}</think>Given functions ( f(x) = cos^2 x ) and ( g(x) = frac{1}{2} + sqrt{3} sin x cos x ):(1) To find the value of ( g(2a) ) where the line ( x = a ) is an axis of symmetry for the graph of ( y = f(x) ):- The function ( f(x) = cos^2 x ) has axes of symmetry at ( x = kpi/2 ) for integers ( k ). This is because ( cos^2 x ) is symmetric about these points.- For such ( a ), ( 2a ) will be an integer multiple of ( pi ). Therefore, ( sin(2a) = 0 ), leading to ( g(2a) = frac{1}{2} + sqrt{3} sin(2a) cos(2a) = frac{1}{2} ).Thus, the value of ( g(2a) ) is (boxed{dfrac{1}{2}}).(2) To find the range of ( h(x) = f(x) + g(x) ) for ( 0 leq x leq frac{pi}{2} ):- Simplify ( h(x) ): [ h(x) = cos^2 x + frac{1}{2} + sqrt{3} sin x cos x ] Using trigonometric identities, this simplifies to: [ h(x) = 1 + cosleft(2x - frac{pi}{3}right) ]- The argument ( 2x - frac{pi}{3} ) ranges from ( -frac{pi}{3} ) to ( frac{2pi}{3} ) as ( x ) goes from 0 to ( frac{pi}{2} ).- The range of ( cos theta ) over ( theta in left[-frac{pi}{3}, frac{2pi}{3}right] ) is ([- frac{1}{2}, 1]). Therefore, the range of ( 1 + costheta ) is ([1 - frac{1}{2}, 1 + 1] = left[frac{1}{2}, 2right]).Thus, the range of ( h(x) ) is (boxed{left[ dfrac{1}{2}, 2 right]}).]

🔑:(1) We know that:f(x) = cos^2{x} = frac{1 + cos{2x}}{2}The axis of symmetry would be at 2x = kpi, where k in mathbb{Z}, thus x = frac{kpi}{2}. Since the line x=a is an axis of symmetry of the graph of y=f(x), we have:a = frac{kpi}{2}, k in mathbb{Z}Given that,g(x) = frac{1}{2} + frac{sqrt{3}}{2}sin{2x}we get:g(2a) = g(kpi) = frac{1}{2} + frac{sqrt{3}}{2}sin{2kpi} = frac{1}{2}since sin{2kpi} = 0 for any k in mathbb{Z}.So, we find that:boxed{g(2a) = frac{1}{2}}(2) From (1), we get:h(x) = f(x) + g(x) = frac{1}{2}cos{2x} + frac{sqrt{3}}{2}sin{2x} + 1By re-writing in the form of a sine function with a phase shift, we can express this as:h(x) = sinleft(2x + frac{pi}{6}right) + 1Since x in [0, frac{pi}{2}], then:2x + frac{pi}{6} in left[frac{pi}{6}, frac{7pi}{6}right]and so:sinleft(2x + frac{pi}{6}right) in left[-frac{1}{2}, 1right]Thus:sinleft(2x + frac{pi}{6}right) + 1 in left[frac{1}{2}, 2right]Therefore, the range of h(x) is:boxed{[ frac{1}{2}, 2 ]}