🧠:[<think>Okay, so I need to find the eccentricity of a hyperbola given by the equation x²/a² - y²/b² = 1, and one of its asymptotes is y = (4/5)x. Hmm, let's recall what I know about hyperbolas. First, the standard form of a hyperbola centered at the origin opening horizontally is indeed x²/a² - y²/b² = 1. The asymptotes of such a hyperbola are the lines y = ±(b/a)x. So, the slopes of the asymptotes are ±b/a. The problem states that one of the asymptotes is y = (4/5)x. Since the hyperbola is symmetric, the other asymptote should be y = -(4/5)x. Therefore, the slope of the asymptotes is 4/5 and -4/5. Comparing this with the general slopes ±b/a, that means that b/a = 4/5. So, b/a = 4/5. Therefore, I can write that b = (4/5)a. Now, the eccentricity e of a hyperbola is given by e = c/a, where c is the distance from the center to each focus. But I also remember that for hyperbolas, the relationship between a, b, and c is c² = a² + b². So, once I find c in terms of a, I can find the eccentricity. Since I know that b = (4/5)a, I can substitute this into the equation for c. Let's compute c²:c² = a² + b²= a² + ( (4/5)a )²= a² + (16/25)a²= (1 + 16/25)a²= (25/25 + 16/25)a²= (41/25)a²Therefore, c = sqrt(41/25)a = (sqrt(41)/5)aTherefore, the eccentricity e = c/a = (sqrt(41)/5)a / a = sqrt(41)/5Wait, but sqrt(41)/5 can be simplified as sqrt(41)/5. Hmm, sqrt(41) is an irrational number, so it can't be simplified further. Therefore, the eccentricity is sqrt(41)/5. Let me check if I did everything correctly.First, the asymptotes slope is 4/5, which is b/a. So b = 4a/5. Then c² = a² + b² = a² + (16a²/25) = (25a² +16a²)/25 = 41a²/25. Therefore, c = (sqrt(41)/5)a. Then e = c/a = sqrt(41)/5. That seems correct.Alternatively, another way to write the eccentricity formula for hyperbola is e = sqrt(1 + (b²/a²)). Let me compute that. Since b/a is 4/5, then (b²/a²) is 16/25. Then e = sqrt(1 + 16/25) = sqrt(41/25) = sqrt(41)/5. Yep, same answer. So that confirms it. Therefore, the eccentricity is sqrt(41)/5. Let me just write that in LaTeX notation. It should be boxed{dfrac{sqrt{41}}{5}}.Final AnswerThe eccentricity of the hyperbola is boxed{dfrac{sqrt{41}}{5}}.</think>The given hyperbola equation is (frac{x^2}{a^2} - frac{y^2}{b^2} = 1). The asymptotes of this hyperbola are (y = pm frac{b}{a}x). We are given one of the asymptotes as (y = frac{4}{5}x), which means (frac{b}{a} = frac{4}{5}). Therefore, (b = frac{4}{5}a).To find the eccentricity (e), we use the relationship (e = frac{c}{a}), where (c) is the distance from the center to each focus. For hyperbolas, (c^2 = a^2 + b^2). Substituting (b = frac{4}{5}a) into this equation:[c^2 = a^2 + left(frac{4}{5}aright)^2][c^2 = a^2 + frac{16}{25}a^2][c^2 = a^2 left(1 + frac{16}{25}right)][c^2 = a^2 left(frac{25}{25} + frac{16}{25}right)][c^2 = frac{41}{25}a^2][c = frac{sqrt{41}}{5}a]Thus, the eccentricity (e) is:[e = frac{c}{a} = frac{frac{sqrt{41}}{5}a}{a} = frac{sqrt{41}}{5}]Therefore, the eccentricity of the hyperbola is (boxed{dfrac{sqrt{41}}{5}}).,<think>Okay, so I need to find the eccentricity of a hyperbola given by the equation x²/a² - y²/b² = 1, and one of its asymptotes is y = (4/5)x. Hmm, eccentricity, right? Let me recall what the formula for eccentricity of a hyperbola is. I think it's e = c/a, where c is the distance from the center to each focus, and a is the distance from the center to each vertex. But wait, I also remember that for hyperbolas, there's a relationship between a, b, and c, which is c² = a² + b². So once I know a and b, I can find c and then the eccentricity. But the problem here is that I don't know a or b, but I do know the equation of one asymptote. The asymptotes for a hyperbola of the form x²/a² - y²/b² = 1 are y = ±(b/a)x. Oh, right! So the slopes of the asymptotes are ±b/a. The given asymptote is y = (4/5)x, so that slope must be equal to b/a. So if the slope is 4/5, then b/a = 4/5. Therefore, b = (4/5)a. That relates b and a. So now, since the hyperbola equation is given in terms of a and b, and we need to find the eccentricity e = c/a. But first, we need to express c in terms of a. From the relationship c² = a² + b², substituting b with (4/5)a gives us c² = a² + (16/25)a². Let me compute that: a² + (16/25)a² = (25/25 + 16/25)a² = (41/25)a². So c² = (41/25)a², which means c = (√41)/5 a. Then the eccentricity e is c/a = (√41)/5. Wait, let me check that again. If c = sqrt(a² + b²) = sqrt(a² + (16/25)a²) = sqrt((25a² + 16a²)/25) = sqrt(41a²/25) = (a sqrt(41))/5. Then dividing by a gives e = sqrt(41)/5. That seems right. But let me verify the steps once more. Given the standard hyperbola equation, the asymptotes are indeed y = ±(b/a)x. Since one of them is 4/5 x, then yes, b/a = 4/5. Therefore, b = (4a)/5. Then c² = a² + b² = a² + (16a²)/25 = (25a² + 16a²)/25 = 41a²/25. Taking the square root, c = (sqrt(41)/5)a. Therefore, e = c/a = sqrt(41)/5. So the eccentricity is sqrt(41)/5. I think that's correct. But just to be thorough, let me make sure there's no missing step. The problem mentions "one of its asymptotes," but since the hyperbola is symmetric, the other asymptote would be y = -4/5 x, which makes sense. The slope is positive and negative. The relation between a, b, and c is standard for hyperbolas. Eccentricity is always greater than 1 for hyperbolas, and sqrt(41)/5 is approximately 6.4/5 = 1.28, which is greater than 1. That seems reasonable. If I compute sqrt(41), it's about 6.403, divided by 5 gives approximately 1.2806, which is a plausible eccentricity. Alternatively, if I made a mistake in the slope, but the given slope is positive 4/5, and since the hyperbola is oriented along the x-axis (since the x² term is positive), the asymptotes should indeed have slopes ±b/a. So that's correct. Therefore, I think the answer is sqrt(41)/5. Just to recap:1. Identify asymptote slope: 4/5 = b/a => b = (4/5)a2. Use hyperbola relationship: c² = a² + b²3. Substitute b: c² = a² + (16/25)a² = (41/25)a² => c = (sqrt(41)/5)a4. Eccentricity e = c/a = sqrt(41)/5All steps check out, so I think that's the correct answer.Final AnswerThe eccentricity of the hyperbola is boxed{dfrac{sqrt{41}}{5}}.</think>To determine the eccentricity of the hyperbola given by the equation (frac{x^2}{a^2} - frac{y^2}{b^2} = 1) with an asymptote (y = frac{4}{5}x), we start by noting that the slopes of the asymptotes of a hyperbola in this form are (pm frac{b}{a}). Given the slope (frac{4}{5}), we have:[frac{b}{a} = frac{4}{5} implies b = frac{4}{5}a]Next, we use the relationship between (a), (b), and (c) (the distance from the center to each focus) for hyperbolas, which is (c^2 = a^2 + b^2). Substituting (b = frac{4}{5}a) into this equation, we get:[c^2 = a^2 + left(frac{4}{5}aright)^2 = a^2 + frac{16}{25}a^2 = frac{25a^2 + 16a^2}{25} = frac{41a^2}{25}]Taking the square root of both sides, we find:[c = frac{sqrt{41}}{5}a]The eccentricity (e) is given by (e = frac{c}{a}). Substituting the value of (c), we get:[e = frac{frac{sqrt{41}}{5}a}{a} = frac{sqrt{41}}{5}]Thus, the eccentricity of the hyperbola is (boxed{dfrac{sqrt{41}}{5}}).]

🔑:To find the eccentricity of the hyperbola, we first need to know the relationship between the asymptote and the parameters a and b. The equation of an asymptote for a hyperbola of the form frac{x^2}{a^2} - frac{y^2}{b^2} = 1 is y = pm frac{b}{a}x.Given the asymptote y = frac{4}{5}x, we can identify that frac{b}{a} = frac{4}{5}. To find the eccentricity, we use the definition that relates the semi-major axis a, the semi-minor axis b, and the distance from the center to the foci c, given by c^2 = a^2 + b^2. Substituting frac{b}{a} into the equation, we get:frac{c^2}{a^2} = frac{b^2}{a^2} + 1 = left(frac{4}{5}right)^2 + 1 = frac{16}{25} + frac{25}{25} = frac{41}{25}.Taking the square root of both sides to find the ratio frac{c}{a}, which gives us the eccentricity e,e = frac{c}{a} = sqrt{frac{41}{25}} = frac{sqrt{41}}{5}.Therefore, the eccentricity of the hyperbola is boxed{frac{sqrt{41}}{5}}.

🧠:[<think>Alright, let's tackle this problem step by step. It's about a regular hexagon and some internal division points on its diagonals. The goal is to find the value of λ such that points B, M, and N are collinear. Hmm, okay. Let me try to visualize the regular hexagon first. So, a regular hexagon ABCDEF, right? Let me draw it mentally. Starting with point A, then B, C, D, E, F, and back to A. All sides are equal, and all internal angles are 120 degrees each. The diagonals mentioned are AC and CE. Wait, in a regular hexagon, the diagonals can have different lengths. For example, AC is a longer diagonal that skips one vertex, while CE would be another diagonal. Let me confirm: in a regular hexagon, the diagonals that connect vertices with two vertices in between are longer. So, AC is one of those longer diagonals, and CE? Let's see. From C to E is also two vertices apart (C to D to E), so CE is also a longer diagonal, same length as AC. Okay, so both AC and CE are of equal length.Now, M and N are internal division points of diagonals AC and CE, respectively. The problem states that AM / AC = CN / CE = λ. Since AC and CE are equal in length, this ratio λ is the same for both. So, M divides AC such that AM = λ * AC, and N divides CE such that CN = λ * CE. Since they are internal division points, λ must be between 0 and 1.We need to find the value of λ such that points B, M, and N are collinear. That means when we connect points B, M, and N, they lie on a straight line. To solve this, I think coordinate geometry might be helpful. If I can assign coordinates to the hexagon's vertices, then find the coordinates of M and N in terms of λ, and then enforce the collinearity condition (slopes between B and M, and between M and N must be equal), that should give me an equation to solve for λ.Let me set up a coordinate system. Let's place the regular hexagon in the plane with its center at the origin. Since regular hexagons can be inscribed in a circle, all vertices will lie on a circle with radius equal to the distance from the center to any vertex. Let's assume the hexagon has a side length of 1 for simplicity. Then, the radius of the circumscribed circle is also 1.Coordinates of the vertices can be determined using trigonometric functions. Starting from vertex A at (1, 0), each subsequent vertex is obtained by rotating 60 degrees around the origin. Let's list the coordinates:- Point A: (1, 0)- Point B: (cos(60°), sin(60°)) = (0.5, √3/2)- Point C: (cos(120°), sin(120°)) = (-0.5, √3/2)- Point D: (cos(180°), sin(180°)) = (-1, 0)- Point E: (cos(240°), sin(240°)) = (-0.5, -√3/2)- Point F: (cos(300°), sin(300°)) = (0.5, -√3/2)Wait, let me confirm these coordinates. Starting from A at (1, 0), moving 60 degrees each time. Point B would be 60 degrees from the x-axis, so coordinates (cos60°, sin60°), which is (0.5, √3/2). Then point C is 120 degrees from the x-axis, so cos120° is cos(180°−60°) = -cos60° = -0.5, sin120° is sin(180°−60°) = sin60° = √3/2. So C is (-0.5, √3/2). Similarly, D is at 180°, which is (-1, 0). E is 240°, which is equivalent to -120°, so cos240° = cos(-120°) = cos120° = -0.5, sin240° = sin(-120°) = -sin60° = -√3/2. So E is (-0.5, -√3/2). F is 300°, which is -60°, so cos300° = 0.5, sin300° = -√3/2. So F is (0.5, -√3/2). Yes, that seems correct.Now, diagonals AC and CE. Let's find coordinates of points M and N.First, diagonal AC. Points A (1, 0) and C (-0.5, √3/2). The vector from A to C is (-1.5, √3/2). So point M divides AC in the ratio AM / AC = λ. Wait, the internal division. So the coordinates of M can be found using the section formula. If a point divides the line segment joining (x1, y1) and (x2, y2) in the ratio m:n, then the coordinates are ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)).But here, AM / AC = λ. Since AC is the entire length, and AM is a part of it. So, the ratio AM : MC would be λ : (1 - λ). Because AM = λ * AC, so MC = AC - AM = (1 - λ) * AC. Therefore, M divides AC in the ratio AM : MC = λ : (1 - λ). So using the section formula, coordinates of M would be:M_x = (λ * C_x + (1 - λ) * A_x) / (λ + (1 - λ)) = λ*(-0.5) + (1 - λ)*1Similarly, M_y = λ*(√3/2) + (1 - λ)*0Simplify M_x:M_x = -0.5λ + 1 - λ = 1 - 1.5λM_y = (√3/2)λSimilarly, for point N on diagonal CE. Diagonal CE connects points C (-0.5, √3/2) and E (-0.5, -√3/2). Wait, is that right? Let me check. Coordinates of C are (-0.5, √3/2), and E is (-0.5, -√3/2). So CE is a vertical line segment from (-0.5, √3/2) to (-0.5, -√3/2). Because both points have the same x-coordinate, so CE is vertical. The problem states that CN / CE = λ. So N is a point on CE such that CN = λ * CE. Since CE is a vertical line, going from C down to E. So starting at C, moving down λ fraction of CE's length to reach N. So the coordinates of N would be:Since CE is from C (-0.5, √3/2) to E (-0.5, -√3/2). The y-coordinate decreases from √3/2 to -√3/2, a total change of -√3. So moving down by λ * √3. Therefore, the coordinates of N would be:N_x = -0.5 (same as C and E)N_y = √3/2 - λ * √3Alternatively, using the section formula. Since CN / CE = λ, which means CN : NE = λ : (1 - λ). So N divides CE in the ratio CN : NE = λ : (1 - λ). Therefore, coordinates of N are:N_x = (λ * E_x + (1 - λ) * C_x) / (λ + (1 - λ)) = λ*(-0.5) + (1 - λ)*(-0.5) = -0.5(λ + 1 - λ) = -0.5N_y = (λ * E_y + (1 - λ) * C_y) / 1 = λ*(-√3/2) + (1 - λ)*(√3/2) = (√3/2)( -λ + 1 - λ ) = (√3/2)(1 - 2λ)So yes, that's the same as before: N_y = √3/2 - λ√3. Because √3/2 - λ√3 = (√3/2)(1 - 2λ). Exactly. So N is at (-0.5, (√3/2)(1 - 2λ)).Alright, so coordinates of M are (1 - 1.5λ, (√3/2)λ), and coordinates of N are (-0.5, (√3/2)(1 - 2λ)).Now, we need points B, M, N to be collinear. Let's recall that point B is at (0.5, √3/2). So three points: B (0.5, √3/2), M (1 - 1.5λ, (√3/2)λ), and N (-0.5, (√3/2)(1 - 2λ)) must lie on a straight line.To check collinearity, the slope between B and M should be equal to the slope between M and N. Alternatively, the area of the triangle formed by B, M, N should be zero. Let's use the slope method.First, compute the slope between B and M.Slope BM = [M_y - B_y] / [M_x - B_x] = [( (√3/2)λ - √3/2 ) / (1 - 1.5λ - 0.5 )]Simplify numerator: (√3/2)(λ - 1)Denominator: 1 - 0.5 - 1.5λ = 0.5 - 1.5λSo slope BM = (√3/2)(λ - 1) / (0.5 - 1.5λ)Similarly, compute slope MN.Slope MN = [N_y - M_y] / [N_x - M_x] = [ ( (√3/2)(1 - 2λ) - (√3/2)λ ) / ( -0.5 - (1 - 1.5λ) ) ]Simplify numerator:(√3/2)(1 - 2λ - λ) = (√3/2)(1 - 3λ)Denominator:-0.5 - 1 + 1.5λ = -1.5 + 1.5λ = 1.5(λ - 1)So slope MN = (√3/2)(1 - 3λ) / [1.5(λ - 1)]Simplify denominator: 1.5 is 3/2, so denominator is (3/2)(λ - 1)Thus, slope MN = (√3/2)(1 - 3λ) / ( (3/2)(λ - 1) ) = (√3/2) * (1 - 3λ) * (2/3) / (λ - 1) )Simplify: (√3/2 * 2/3) = √3/3. So:Slope MN = (√3/3) * (1 - 3λ) / (λ - 1) = (√3/3) * [ - (3λ - 1) / (λ - 1) ) ] = (√3/3) * [ (3λ - 1) / (1 - λ) ) ]Wait, let me check again:(1 - 3λ) / (λ - 1) = - (3λ - 1) / (λ - 1) = (3λ - 1) / (1 - λ)Yes, so slope MN = (√3/3) * (3λ - 1)/(1 - λ)Now, set slope BM equal to slope MN.Slope BM: (√3/2)(λ - 1)/(0.5 - 1.5λ)Slope MN: (√3/3)(3λ - 1)/(1 - λ)Set them equal:(√3/2)(λ - 1)/(0.5 - 1.5λ) = (√3/3)(3λ - 1)/(1 - λ)We can cancel √3 from both sides:(1/2)(λ - 1)/(0.5 - 1.5λ) = (1/3)(3λ - 1)/(1 - λ)Note that (λ - 1) = - (1 - λ), and (0.5 - 1.5λ) = -1.5λ + 0.5 = - (1.5λ - 0.5). Let's rewrite:Left side: (1/2)*(- (1 - λ)) / (- (1.5λ - 0.5)) = (1/2)*(1 - λ)/(1.5λ - 0.5)Right side: (1/3)(3λ - 1)/(1 - λ)So equation becomes:(1/2)*(1 - λ)/(1.5λ - 0.5) = (1/3)*(3λ - 1)/(1 - λ)Multiply both sides by (1.5λ - 0.5)(1 - λ) to eliminate denominators:(1/2)*(1 - λ)^2 = (1/3)*(3λ - 1)(1.5λ - 0.5)Multiply both sides by 6 to eliminate fractions:3*(1 - λ)^2 = 2*(3λ - 1)(1.5λ - 0.5)Let's compute each side.Left side: 3*(1 - 2λ + λ²) = 3 - 6λ + 3λ²Right side: 2*(3λ - 1)(1.5λ - 0.5). Let's compute (3λ - 1)(1.5λ - 0.5):First, multiply 3λ * 1.5λ = 4.5λ²3λ * (-0.5) = -1.5λ-1 * 1.5λ = -1.5λ-1 * (-0.5) = 0.5So total: 4.5λ² - 1.5λ -1.5λ + 0.5 = 4.5λ² - 3λ + 0.5Multiply by 2: 9λ² - 6λ + 1Thus, right side is 9λ² -6λ +1Set left and right sides equal:3 -6λ +3λ² =9λ² -6λ +1Subtract left side from both sides:0 =9λ² -6λ +1 -3 +6λ -3λ²Simplify:9λ² -3λ² =6λ²-6λ +6λ =01 -3 =-2Thus: 6λ² -2=0Therefore:6λ²=2 => λ²=1/3 => λ=±1/√3But since λ is a ratio between 0 and1 (since it's an internal division), we take the positive root: λ=1/√3But the problem might expect a rationalized denominator, so λ=√3/3Hence, the value of λ is √3/3.But let me check the calculations again to be sure.So equation after simplifying slopes led to 6λ² -2=0, so λ=√(1/3)=√3/3. Yes, that seems correct.Alternatively, let's cross-verify by plugging λ=√3/3 into coordinates and check collinearity.Compute coordinates of M:M_x =1 -1.5*(√3/3)=1 - (1.5/3)*√3=1 -0.5√3≈1 -0.866≈0.134M_y= (√3/2)*(√3/3)= (3/2)/3=0.5So M≈(0.134, 0.5)Coordinates of N:N_x=-0.5N_y=(√3/2)(1 -2*(√3/3))= (√3/2)(1 -2√3/3)= (√3/2)(3 -2√3)/3= (√3*(3 -2√3))/6Let's compute that:√3*(3 -2√3)/6= (3√3 -6)/6= (3√3)/6 -6/6= (√3/2) -1≈0.866/2 -1≈0.433 -1≈-0.567So N≈(-0.5, -0.567)Point B is at (0.5, √3/2≈0.866)So points B(0.5,0.866), M(0.134,0.5), N(-0.5,-0.567). Let's check if these are collinear.Compute the slope from B to M: (0.5 -0.866)/(0.134 -0.5)= (-0.366)/(-0.366)=1Slope from M to N: (-0.567 -0.5)/(-0.5 -0.134)= (-1.067)/(-0.634)≈1.683Wait, that's inconsistent. Hmm, so according to this, slopes are 1 and ~1.683, which are not equal. That suggests an error in my calculation.Wait, but according to the algebra, λ=√3/3≈0.577. Wait, when I plugged in λ=√3/3≈0.577, let me recalculate the coordinates.First, M_x=1 -1.5λ=1 -1.5*(√3/3)=1 - (√3/2)≈1 -0.866≈0.134 (correct). M_y=(√3/2)*λ=(√3/2)*(√3/3)= (3/2)/3=0.5 (correct).N_y=(√3/2)(1 -2λ)= (√3/2)(1 -2*(√3/3))= (√3/2)(1 - (2√3)/3 )Compute 1 - (2√3)/3 ≈1 - (3.464)/3≈1 -1.154≈-0.154. So N_y=(√3/2)*(-0.154)≈0.866*(-0.154)/2≈-0.066. Wait, no, wait. Wait, my previous calculation was wrong. Let me do this again:N_y=(√3/2)(1 -2λ)= (√3/2)(1 -2*(√3/3))= (√3/2)*(1 - (2√3)/3 )Let me compute 1 - (2√3)/3:Multiply numerator and denominator by 3: (3 -2√3)/3Therefore, N_y= (√3/2)*( (3 -2√3)/3 )= (√3*(3 -2√3))/6 = (3√3 - 6)/6= (3√3)/6 -6/6= (√3/2) -1≈0.866/2 -1≈0.433 -1≈-0.567. Wait, so why did I get different results before? Because (√3/2)(1 -2λ)= (√3/2)*(1 -2*(√3/3)) is indeed (√3/2)*(1 - 2√3/3). Wait, 2√3/3 is approximately 1.154, so 1 -1.154≈-0.154, then multiply by √3/2≈0.866/2≈0.433, so 0.433*(-0.154)≈-0.066. Wait, but this contradicts the previous calculation.Wait, there must be an error in the algebra here. Let me redo the computation of N_y.Given λ=√3/3≈0.577.N divides CE such that CN / CE=λ. CE is the length from C to E. Since in our coordinate system, CE is a vertical line from (-0.5, √3/2) to (-0.5, -√3/2). The length CE is the distance between these two points, which is |√3/2 - (-√3/2)|=√3. So CE has length √3. Therefore, CN=λ*CE= (√3/3)*√3=1. So CN=1. But CE is length √3, so moving 1 unit from C towards E. But wait, that seems inconsistent because the total length is √3≈1.732, so moving 1 unit from C would not reach E. Wait, but in coordinates, from C at (-0.5, √3/2≈0.866) moving down 1 unit along CE (which is vertical). The y-coordinate of C is √3/2≈0.866, so moving down 1 unit would take us to y≈0.866 -1≈-0.134. But E is at (-0.5, -√3/2≈-0.866). So perhaps there's confusion here between actual lengths and coordinate distances.Wait a second, in coordinate terms, the distance from C to E is sqrt( (0)^2 + (-√3/2 - √3/2)^2 )=sqrt( (-√3)^2 )=sqrt(3). So CE has length sqrt(3). Therefore, CN=λ*sqrt(3). But in coordinates, the y-coordinate changes from √3/2 to -√3/2. So the total change in y is -√3. Therefore, moving down by λ*sqrt(3) in terms of actual length. But how does this translate to coordinate displacement? Since the y-component is moving down by sqrt(3)*λ. So the coordinates of N would be:N_x= -0.5 (same as C and E)N_y= (√3/2) - (sqrt(3)*λ)But in the problem statement, it's given that CN / CE=λ. Since CE is sqrt(3), then CN=λ*sqrt(3). Therefore, the y-coordinate decreases by CN=λ*sqrt(3), so N_y= C_y - CN= (√3/2) - λ*sqrt(3). Which is the same as N_y= sqrt(3)/2 - sqrt(3)*lambda= sqrt(3)*(1/2 - lambda). Wait, but in the previous calculations using section formula, we had N_y=(sqrt(3)/2)(1 - 2lambda). Hmm. Let's check.Wait, if CE is from C(-0.5, sqrt(3)/2) to E(-0.5, -sqrt(3)/2). The vector from C to E is (0, -sqrt(3)). So the parametric equation for CE is (-0.5, sqrt(3)/2 - t*sqrt(3)), where t ranges from 0 to 1. So when t=0, we are at C, and t=1, at E. Therefore, for a general t, N would be at (-0.5, sqrt(3)/2 - t*sqrt(3)). The length CN is t*sqrt(3), so CN / CE= t*sqrt(3)/sqrt(3)=t. Therefore, t=lambda. Therefore, N is at (-0.5, sqrt(3)/2 - lambda*sqrt(3)). Therefore, N_y= sqrt(3)/2 - sqrt(3)*lambda= sqrt(3)(1/2 - lambda). But in the section formula approach earlier, we had N_y=(sqrt(3)/2)(1 - 2lambda). Wait, these two should be the same. Let's check:sqrt(3)(1/2 - lambda) versus (sqrt(3)/2)(1 - 2lambda). Indeed, sqrt(3)(1/2 - lambda)= sqrt(3)/2 - sqrt(3)lambda= (sqrt(3)/2)(1 - 2lambda). Yes, they are equal. So both expressions for N_y are consistent. Therefore, N_y= sqrt(3)/2 - sqrt(3)lambda.Therefore, plugging lambda=√3/3 into N_y:N_y= sqrt(3)/2 - sqrt(3)*(√3/3)= sqrt(3)/2 - (3/3)= sqrt(3)/2 -1≈0.866 -1≈-0.134Wait, this contradicts my previous calculation where I thought N_y≈-0.567. Wait, so where did that mistake happen?Earlier, when I calculated N_y=(sqrt(3)/2)(1 - 2lambda) with lambda=√3/3, that becomes (sqrt(3)/2)(1 - 2*(√3/3))=(sqrt(3)/2)(1 - (2sqrt(3))/3 ). Let's compute 1 - (2sqrt(3))/3 numerically: 2sqrt(3)=3.464, so 3.464/3≈1.154, so 1 -1.154≈-0.154. Then multiply by sqrt(3)/2≈0.866/2≈0.433. So 0.433*(-0.154)≈-0.066. But according to the parametric expression, N_y= sqrt(3)/2 - sqrt(3)*lambda= sqrt(3)*(1/2 - lambda)= sqrt(3)*(0.5 - 0.577)= sqrt(3)*(-0.077)≈1.732*(-0.077)≈-0.134. Wait, but these two results are different. Which is correct?Wait, there's confusion here. Let's compute N_y using both expressions:First, with lambda=√3/3≈0.577.Using N_y= sqrt(3)/2 - sqrt(3)*lambda= sqrt(3)/2 - sqrt(3)*(sqrt(3)/3)= sqrt(3)/2 - (3)/3= sqrt(3)/2 -1≈0.866 -1≈-0.134Using the section formula: N_y=(sqrt(3)/2)(1 - 2lambda)= (sqrt(3)/2)(1 - 2*(sqrt(3)/3))= (sqrt(3)/2)(1 - (2sqrt(3))/3 )Compute 1 - (2sqrt(3))/3≈1 -1.154≈-0.154Multiply by sqrt(3)/2≈0.866/2≈0.4330.433*(-0.154)≈-0.066Wait, these two results don't match. There must be a miscalculation. Wait, sqrt(3)/2 is approximately 0.866/2=0.433? Wait no, sqrt(3)≈1.732, so sqrt(3)/2≈0.866. So (sqrt(3)/2)*(1 - 2lambda)=0.866*(1 -2*0.577)=0.866*(1 -1.154)=0.866*(-0.154)≈-0.134. Ah, there we go. I think I made a mistake in the intermediate step.Wait, sqrt(3)/2≈0.866. Then, 1 -2lambda≈1 -2*(0.577)=1 -1.154≈-0.154. Then, 0.866*(-0.154)≈-0.134. So both expressions give N_y≈-0.134. Therefore, correct coordinates for N are (-0.5, -0.134). Not -0.567. My earlier mistake was incorrectly expanding the algebraic expression. So point N is at (-0.5, approximately -0.134).So points B(0.5,0.866), M≈(0.134,0.5), N≈(-0.5,-0.134). Let's check if these are collinear.Compute the slope from B to M: (0.5 -0.866)/(0.134 -0.5)= (-0.366)/(-0.366)=1.0Slope from M to N: (-0.134 -0.5)/(-0.5 -0.134)= (-0.634)/(-0.634)=1.0Ah, so slopes are both 1.0, which means collinear. So that checks out. My previous mistake was miscalculating N_y due to arithmetic errors. So with λ=√3/3≈0.577, the points are collinear. Therefore, the value of λ is √3/3.Therefore, the answer is boxed{dfrac{sqrt{3}}{3}}.Final Answerboxed{dfrac{sqrt{3}}{3}}</think>To solve the problem, we start by placing the regular hexagon (ABCDEF) in a coordinate system with its center at the origin. The vertices of the hexagon are determined using trigonometric functions. The coordinates of the vertices are:- (A(1, 0))- (Bleft(frac{1}{2}, frac{sqrt{3}}{2}right))- (Cleft(-frac{1}{2}, frac{sqrt{3}}{2}right))- (D(-1, 0))- (Eleft(-frac{1}{2}, -frac{sqrt{3}}{2}right))- (Fleft(frac{1}{2}, -frac{sqrt{3}}{2}right))Next, we find the coordinates of points (M) and (N) which divide the diagonals (AC) and (CE) respectively in the ratio (lambda). For point (M) on (AC):- The coordinates of (M) are calculated using the section formula: [ Mleft(1 - 1.5lambda, frac{sqrt{3}}{2}lambdaright) ]For point (N) on (CE):- The coordinates of (N) are calculated using the section formula: [ Nleft(-0.5, frac{sqrt{3}}{2}(1 - 2lambda)right) ]To ensure collinearity of points (B), (M), and (N), we check if the slopes between (B) and (M), and between (M) and (N) are equal. The slope between (B) and (M) is:[text{slope}_{BM} = frac{frac{sqrt{3}}{2}lambda - frac{sqrt{3}}{2}}{1 - 1.5lambda - 0.5} = frac{frac{sqrt{3}}{2}(lambda - 1)}{0.5 - 1.5lambda}]The slope between (M) and (N) is:[text{slope}_{MN} = frac{frac{sqrt{3}}{2}(1 - 2lambda) - frac{sqrt{3}}{2}lambda}{-0.5 - (1 - 1.5lambda)} = frac{frac{sqrt{3}}{2}(1 - 3lambda)}{-1.5 + 1.5lambda}]Setting these slopes equal and solving for (lambda), we find:[frac{frac{sqrt{3}}{2}(lambda - 1)}{0.5 - 1.5lambda} = frac{frac{sqrt{3}}{3}(3lambda - 1)}{1 - lambda}]Simplifying this equation, we find:[6lambda^2 - 2 = 0 implies lambda^2 = frac{1}{3} implies lambda = frac{sqrt{3}}{3}]Thus, the value of (lambda) is (boxed{dfrac{sqrt{3}}{3}}).,<think>Okay, so I have this problem about a regular hexagon named ABCDEF. The problem mentions points M and N, which are internal division points of the diagonals AC and CE respectively. The ratio of AM to AC is equal to the ratio of CN to CE, both being lambda (λ). The key condition here is that points B, M, and N are collinear. My task is to find the value of λ.First, let me try to visualize the regular hexagon. In a regular hexagon, all sides are equal, and all internal angles are 120 degrees. The diagonals AC and CE are mentioned. Let me recall that in a regular hexagon, the diagonals can be of different lengths. For example, the shorter diagonals (connecting two vertices with one vertex in between) are equal in length to the side length multiplied by √3, while the longer diagonals (connecting opposite vertices) are twice the side length. Wait, maybe I should confirm that.Actually, in a regular hexagon with side length 's', the length of the diagonal AC (connecting A to C) would be 2s times sin(60°) because each internal angle is 120°, but the angle between adjacent vertices from the center is 60°. Hmm, perhaps I should use coordinates to model this problem. That might make it easier to handle the collinearity condition.Yes, coordinate geometry might be a good approach here. Let me assign coordinates to the hexagon. Let's place the hexagon in the coordinate plane with its center at the origin (0,0). Let's assume the hexagon has a side length of 1 for simplicity. Then, the vertices can be determined using polar coordinates converted to Cartesian.In a regular hexagon, each vertex is separated by 60° (or π/3 radians). Starting from vertex A at (1,0), the coordinates of the vertices would be:- A: (1, 0)- B: (cos(60°), sin(60°)) = (0.5, √3/2)- C: (-0.5, √3/2)- D: (-1, 0)- E: (-0.5, -√3/2)- F: (0.5, -√3/2)Wait, let me verify that. Starting from the rightmost point A at (1,0), moving counterclockwise by 60°, each subsequent vertex. So point B is at 60°, which is (cos 60°, sin 60°) = (0.5, √3/2). Then point C is at 120°, which is (-0.5, √3/2). Then D is at 180°, which is (-1,0). E is at 240°, which is (-0.5, -√3/2). F is at 300°, which is (0.5, -√3/2), and back to A at 360°=0°.So coordinates are correct. Now, diagonals AC and CE. Diagonal AC connects points A(1,0) to C(-0.5, √3/2). Diagonal CE connects point C(-0.5, √3/2) to E(-0.5, -√3/2). Wait, CE is from C to E. Let me check the coordinates of E: (-0.5, -√3/2). So CE is a vertical line since the x-coordinate is the same (-0.5), and the y-coordinate goes from √3/2 to -√3/2. So CE is a vertical diagonal of length √3.Similarly, AC connects A(1,0) to C(-0.5, √3/2). Let me compute the coordinates of M and N. Since M is an internal division point of AC with AM/AC = λ, and N is an internal division point of CE with CN/CE = λ.So for point M on AC: starting at A(1,0) and moving towards C(-0.5, √3/2), the vector from A to C is (-1.5, √3/2). So the coordinates of M can be expressed as A + λ*(vector AC). So M = (1,0) + λ*(-1.5, √3/2) = (1 - 1.5λ, 0 + (√3/2)λ) = (1 - 1.5λ, (√3/2)λ).Similarly, point N is on CE with CN/CE = λ. Since CE is from C(-0.5, √3/2) to E(-0.5, -√3/2). So the vector from C to E is (0, -√3). Therefore, since CN/CE = λ, the coordinates of N are C + (1 - λ)*(vector CE). Wait, wait: if CN/CE = λ, does that mean N divides CE internally in the ratio CN:NE = λ:(1 - λ)? So starting at C, moving towards E by λ fraction of CE. So the coordinates of N would be C + λ*(vector CE). The vector CE is (0, -√3). Therefore, N = (-0.5, √3/2) + λ*(0, -√3) = (-0.5, √3/2 - √3*λ).Therefore, coordinates:M: (1 - 1.5λ, (√3/2)λ)N: (-0.5, √3/2 - √3*λ)Now, we need points B, M, N to be collinear. So the three points B, M, N should lie on the same straight line. To check collinearity, we can use the area of the triangle formed by these three points: if the area is zero, they are collinear. Alternatively, we can compute the slopes between B and M, and between B and N, and set them equal.Let me compute the coordinates of B: (0.5, √3/2). So point B is at (0.5, √3/2).So the three points are:B: (0.5, √3/2)M: (1 - 1.5λ, (√3/2)λ)N: (-0.5, √3/2 - √3λ)We can use the slope method. Let's compute the slope from B to M and from B to N. If they are equal, then the three points are collinear.First, compute the slope BM:Slope BM = [y_M - y_B] / [x_M - x_B] = [( (√3/2)λ - √3/2 ) / ( (1 - 1.5λ) - 0.5 )]Simplify numerator: (√3/2)(λ - 1)Denominator: (1 - 1.5λ - 0.5) = (0.5 - 1.5λ)So slope BM = [ (√3/2)(λ - 1) ] / (0.5 - 1.5λ )Similarly, compute slope BN:Slope BN = [y_N - y_B] / [x_N - x_B] = [ ( (√3/2 - √3λ) - √3/2 ) / ( (-0.5 - 0.5 ) ) ]Simplify numerator: (√3/2 - √3λ - √3/2 ) = -√3λDenominator: (-1)Thus, slope BN = (-√3λ) / (-1) = √3λTherefore, setting slope BM = slope BN:[ (√3/2)(λ - 1) ] / (0.5 - 1.5λ ) = √3λWe can divide both sides by √3 to simplify:[ (1/2)(λ - 1) ] / (0.5 - 1.5λ ) = λMultiply numerator and denominator by 2 to eliminate fractions:[ (λ - 1) ] / (1 - 3λ ) = λSo:(λ - 1) = λ(1 - 3λ )Expand the right-hand side:λ - 1 = λ - 3λ²Subtract λ from both sides:-1 = -3λ²Multiply both sides by -1:1 = 3λ²Thus, λ² = 1/3Therefore, λ = ±1/√3But since M and N are internal division points, λ must be between 0 and 1. Hence, λ = 1/√3. However, we should rationalize the denominator, so λ = √3/3.Wait, but let me verify that step where I divided both sides by √3. Let's check the algebra again to be sure.Original equation after slopes:[ (√3/2)(λ - 1) ] / (0.5 - 1.5λ ) = √3λMultiply both sides by (0.5 - 1.5λ ):(√3/2)(λ - 1) = √3λ (0.5 - 1.5λ )Divide both sides by √3:(1/2)(λ - 1) = λ (0.5 - 1.5λ )Multiply out the right-hand side:1/2 λ - 1/2 = 0.5λ - 1.5λ²Subtract 0.5λ from both sides:-1/2 = -1.5λ²Multiply both sides by -1:1/2 = 1.5λ²Divide both sides by 1.5:(1/2) / (3/2) = λ²Which is (1/2)*(2/3) = λ² => 1/3 = λ²Thus, λ = ±1/√3. As before, since it's an internal division, λ must be positive and between 0 and 1, so λ = 1/√3 which is approximately 0.577. Rationalizing, that's √3/3 ≈ 0.577. So λ = √3/3.Wait, but the problem says "internal division points", so division ratio λ must be between 0 and 1. Therefore, the correct value is √3/3.But let me cross-verify using another method. Maybe using vectors or parametric equations.Alternatively, parametrize the line BMN and check where N lies.Let me find the parametric equation of line BM. Since points B, M, N are collinear, N must lie on line BM.So parametric equations: starting at B(0.5, √3/2), moving towards M(1 - 1.5λ, (√3/2)λ). The direction vector from B to M is (1 - 1.5λ - 0.5, (√3/2)λ - √3/2) = (0.5 - 1.5λ, (√3/2)(λ - 1)).So parametric equations:x = 0.5 + t*(0.5 - 1.5λ)y = √3/2 + t*( (√3/2)(λ - 1) )We need this line to pass through N(-0.5, √3/2 - √3λ). So set up equations:For x-coordinate:0.5 + t*(0.5 - 1.5λ) = -0.5Solve for t:t*(0.5 - 1.5λ) = -1t = -1 / (0.5 - 1.5λ) = 1 / (1.5λ - 0.5) = (2/3) / (λ - 1/3) [by multiplying numerator and denominator by 2]Wait, let's compute t:t = (-1) / (0.5 - 1.5λ) = 1 / (1.5λ - 0.5)Similarly, for y-coordinate:√3/2 + t*( (√3/2)(λ - 1) ) = √3/2 - √3λSubtract √3/2 from both sides:t*( (√3/2)(λ - 1) ) = -√3λDivide both sides by √3/2:t*(λ - 1) = -2λSo, from the x-coordinate, t = 1 / (1.5λ - 0.5). Substitute this into the y-coordinate equation:[1 / (1.5λ - 0.5)]*(λ - 1) = -2λMultiply both sides by (1.5λ - 0.5):(λ - 1) = -2λ*(1.5λ - 0.5)Simplify RHS:-2λ*(1.5λ - 0.5) = -3λ² + λSo, equation becomes:λ - 1 = -3λ² + λSubtract λ from both sides:-1 = -3λ²Multiply both sides by -1:1 = 3λ²Thus, λ² = 1/3λ = √(1/3) = 1/√3 = √3/3 ≈ 0.577, which is the same as before. So this confirms the result.Alternatively, maybe using barycentric coordinates or mass point geometry, but given that coordinate geometry worked and the answer is consistent, I think this is solid.Another sanity check: since in a regular hexagon, certain symmetries might be at play. The value λ=√3/3 is approximately 0.577, which is less than 1, so it's a valid internal division.Alternatively, considering vectors:Let me assign vectors to the points. Let’s set point A at (1,0), as before.Vectors:A: (1, 0)B: (0.5, √3/2)C: (-0.5, √3/2)E: (-0.5, -√3/2)So vector AC = C - A = (-1.5, √3/2)So point M divides AC such that AM = λ AC. So vector AM = λ*(-1.5, √3/2), hence M = A + vector AM = (1,0) + (-1.5λ, (√3/2)λ) = (1 - 1.5λ, (√3/2)λ) as before.Similarly, vector CE = E - C = (0, -√3)Point N divides CE such that CN = λ CE. So vector CN = λ*(0, -√3), hence N = C + vector CN = (-0.5, √3/2) + (0, -√3 λ) = (-0.5, √3/2 - √3 λ) as before.The line BMN must be straight. Let's parametrize the line BM and check if N is on it.Parametrize BM: from B(0.5, √3/2) to M(1 - 1.5λ, (√3/2)λ). Let parameter be t from 0 to 1.Parametric equations:x(t) = 0.5 + t*(1 - 1.5λ - 0.5) = 0.5 + t*(0.5 - 1.5λ)y(t) = √3/2 + t*( (√3/2)λ - √3/2 ) = √3/2 + t*( (√3/2)(λ - 1) )We need to find t such that x(t) = -0.5 and y(t) = √3/2 - √3 λ.Solve for t from the x(t) equation:0.5 + t*(0.5 - 1.5λ) = -0.5t*(0.5 - 1.5λ) = -1t = -1 / (0.5 - 1.5λ) = 1 / (1.5λ - 0.5)Now plug this t into y(t):√3/2 + [1 / (1.5λ - 0.5)] * ( (√3/2)(λ - 1) ) = √3/2 - √3 λMultiply both sides by (1.5λ - 0.5):√3/2*(1.5λ - 0.5) + (√3/2)(λ - 1) = (√3/2 - √3 λ)(1.5λ - 0.5)Factor out √3/2 on the left side:√3/2 [ (1.5λ - 0.5) + (λ - 1) ] = √3/2 [1.5λ - 0.5 + λ - 1] = √3/2 [2.5λ - 1.5]Right side:√3/2*(1.5λ - 0.5) - √3 λ*(1.5λ - 0.5)Factor out √3:√3 [ (1/2)(1.5λ - 0.5) - λ(1.5λ - 0.5) ]Simplify inside the brackets:(0.75λ - 0.25) - 1.5λ² + 0.5λCombine like terms:0.75λ + 0.5λ - 1.5λ² - 0.25 = 1.25λ - 1.5λ² - 0.25Therefore, right side becomes:√3 ( -1.5λ² + 1.25λ - 0.25 )Setting left and right sides equal:√3/2 (2.5λ - 1.5) = √3 ( -1.5λ² + 1.25λ - 0.25 )Divide both sides by √3:(1/2)(2.5λ - 1.5) = -1.5λ² + 1.25λ - 0.25Multiply left side:1.25λ - 0.75 = -1.5λ² + 1.25λ - 0.25Subtract 1.25λ from both sides:-0.75 = -1.5λ² - 0.25Add 0.25 to both sides:-0.5 = -1.5λ²Multiply both sides by -1:0.5 = 1.5λ²Divide by 1.5:λ² = 0.5 / 1.5 = 1/3Thus, λ = √(1/3) = √3/3, same result as before.So after verifying through two different methods (slopes and parametric equations), the answer is consistent. Therefore, the value of λ is √3/3.Final Answerboxed{dfrac{sqrt{3}}{3}}</think>Given a regular hexagon (ABCDEF) with points (M) and (N) as internal division points of the diagonals (AC) and (CE) respectively, such that (frac{AM}{AC} = frac{CN}{CE} = lambda). We need to find the value of (lambda) given that points (B), (M), and (N) are collinear.1. Coordinates of Hexagon Vertices: - (A(1, 0)) - (Bleft(frac{1}{2}, frac{sqrt{3}}{2}right)) - (Cleft(-frac{1}{2}, frac{sqrt{3}}{2}right)) - (Eleft(-frac{1}{2}, -frac{sqrt{3}}{2}right))2. Coordinates of Points (M) and (N): - (M) divides (AC) in the ratio (lambda): [ M = left(1 - 1.5lambda, frac{sqrt{3}}{2}lambdaright) ] - (N) divides (CE) in the ratio (lambda): [ N = left(-frac{1}{2}, frac{sqrt{3}}{2} - sqrt{3}lambdaright) ]3. Collinearity Condition: - Calculate the slopes of (BM) and (BN) and set them equal: - Slope of (BM): [ frac{left(frac{sqrt{3}}{2}lambda - frac{sqrt{3}}{2}right)}{left(1 - 1.5lambda - 0.5right)} = frac{frac{sqrt{3}}{2}(lambda - 1)}{0.5 - 1.5lambda} ] - Slope of (BN): [ frac{left(frac{sqrt{3}}{2} - sqrt{3}lambda - frac{sqrt{3}}{2}right)}{-1} = sqrt{3}lambda ]4. Equating Slopes: - Simplify and solve for (lambda): [ frac{frac{sqrt{3}}{2}(lambda - 1)}{0.5 - 1.5lambda} = sqrt{3}lambda ] [ frac{lambda - 1}{0.5 - 1.5lambda} = 2lambda ] [ lambda - 1 = 2lambda(0.5 - 1.5lambda) ] [ lambda - 1 = lambda - 3lambda^2 ] [ -1 = -3lambda^2 ] [ lambda^2 = frac{1}{3} ] [ lambda = frac{sqrt{3}}{3} ]Thus, the value of (lambda) is (boxed{dfrac{sqrt{3}}{3}}).]

🔑:Solution: Extend E A and C B to intersect at point P. Let the side length of the regular hexagon be 1, then P B=2, and A is the midpoint of E P, so E A=A P =sqrt{3}.Since boldsymbol{A M}=lambda A C, then C M=(1-lambda) C A.Since C P=3 C B, and C A is the median of triangle P C E on side P E, C N=lambda C E, so,C A=frac{1}{2}(C E+C P)=frac{1}{2 lambda} C N+frac{3}{2} C B .Also, C A=frac{C M}{1-lambda}, hencebegin{array}{l}frac{1}{2 lambda} C N+frac{3}{2} C B=frac{C M}{1-lambda}, C M=frac{1-lambda}{2 lambda} C N+frac{3(1-lambda)}{2} C B .end{array}When B, M, and N are collinear, there exists a real number t such that C M=(1-t) C N+t C B. Therefore,frac{1-lambda}{2 lambda}+frac{3(1-lambda)}{2}=1 text {. }Solving for lambda gives lambda=frac{sqrt{3}}{3}.

🧠:[<think>Alright, so I've got this system of equations here:[begin{cases}x^3 + 3xy^2 = 158 3x^2y + y^3 = -185end{cases}]Hmm, these look kind of similar to the formulas for expanding ((x + y)^3) and ((x - y)^3). Let me recall those:[(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3][(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3]Wait, if I add those two equations together, the cross terms would cancel out. Let me check:[(x + y)^3 + (x - y)^3 = 2x^3 + 6xy^2]Oh! That's exactly twice the first equation given here. Let me see:The first equation is (x^3 + 3xy^2 = 158). If I multiply that by 2, it becomes (2x^3 + 6xy^2 = 316), which matches the sum of ((x + y)^3 + (x - y)^3).Similarly, if I subtract ((x - y)^3) from ((x + y)^3), I get:[(x + y)^3 - (x - y)^3 = 6x^2y + 2y^3]Which is twice the second equation here. The second equation is (3x^2y + y^3 = -185). Multiplying by 2 gives (6x^2y + 2y^3 = -370), which matches the subtraction result.So, this gives me an idea. Let me denote:[A = x + y B = x - y]Then, according to the earlier expansions:[A^3 + B^3 = 2x^3 + 6xy^2 = 2(x^3 + 3xy^2) = 2 times 158 = 316][A^3 - B^3 = 6x^2y + 2y^3 = 2(3x^2y + y^3) = 2 times (-185) = -370]So now, I have a new system of equations in terms of (A^3) and (B^3):[begin{cases}A^3 + B^3 = 316 A^3 - B^3 = -370end{cases}]If I add these two equations together:[2A^3 = 316 + (-370) = -54 implies A^3 = -27 implies A = sqrt[3]{-27} = -3]Subtracting the second equation from the first:[2B^3 = 316 - (-370) = 316 + 370 = 686 implies B^3 = 343 implies B = sqrt[3]{343} = 7]So, now I know that:[A = x + y = -3 B = x - y = 7]Now, I can solve for (x) and (y) by adding and subtracting these two equations.Adding (A) and (B):[(x + y) + (x - y) = -3 + 7 2x = 4 x = 2]Subtracting (B) from (A):[(x + y) - (x - y) = -3 - 7 2y = -10 y = -5]So, the solution is (x = 2) and (y = -5). Let me check these values in the original equations to make sure I didn't make a mistake.First equation: (x^3 + 3xy^2)Plugging in (x = 2) and (y = -5):[2^3 + 3(2)(-5)^2 = 8 + 3(2)(25) = 8 + 150 = 158]Which matches the first equation.Second equation: (3x^2y + y^3)Plugging in the values:[3(2)^2(-5) + (-5)^3 = 3(4)(-5) + (-125) = -60 - 125 = -185]That also matches the second equation. So, the solution seems correct. Therefore, the answer is (x = 2) and (y = -5).Final AnswerThe solution is (boxed{x = 2}) and (boxed{y = -5}).</think>Given the system of equations:[begin{cases}x^3 + 3xy^2 = 158 3x^2y + y^3 = -185end{cases}]We recognize that these equations resemble the expansions of ((x + y)^3) and ((x - y)^3). By denoting (A = x + y) and (B = x - y), we can transform the given equations:1. The first equation is equivalent to (frac{A^3 + B^3}{2} = 158), leading to (A^3 + B^3 = 316).2. The second equation is equivalent to (frac{A^3 - B^3}{2} = -185), leading to (A^3 - B^3 = -370).We solve the new system of equations for (A^3) and (B^3):[begin{cases}A^3 + B^3 = 316 A^3 - B^3 = -370end{cases}]Adding these equations:[2A^3 = -54 implies A^3 = -27 implies A = -3]Subtracting these equations:[2B^3 = 686 implies B^3 = 343 implies B = 7]Thus, we have (A = x + y = -3) and (B = x - y = 7). Solving for (x) and (y):Adding (A) and (B):[(x + y) + (x - y) = -3 + 7 implies 2x = 4 implies x = 2]Subtracting (B) from (A):[(x + y) - (x - y) = -3 - 7 implies 2y = -10 implies y = -5]Verifying the solution in the original equations confirms the values. Therefore, the solution is:[boxed{x = 2} quad text{and} quad boxed{y = -5}],<think>Alright, let me try to tackle this system of equations. So, we have:1. ( x^{3} + 3x y^{2} = 158 )2. ( 3x^{2} y + y^{3} = -185 )Hmm, these look kind of similar to the expansions of ( (x + y)^3 ) and ( (x - y)^3 ). Let me recall that:( (x + y)^3 = x^3 + 3x^2 y + 3x y^2 + y^3 )and( (x - y)^3 = x^3 - 3x^2 y + 3x y^2 - y^3 )Comparing these with the given equations, if I subtract the second equation from the first, what do I get?First equation: ( x^3 + 3x y^2 = 158 )Second equation: ( 3x^2 y + y^3 = -185 )If I add them together, it would be ( x^3 + 3x y^2 + 3x^2 y + y^3 = 158 + (-185) = -27 )Wait, that's ( (x + y)^3 = -27 ). Because ( (x + y)^3 = x^3 + 3x^2 y + 3x y^2 + y^3 ). Yes, exactly! So adding the two equations gives ( (x + y)^3 = -27 ). Therefore, ( x + y = sqrt[3]{-27} = -3 ). So we have ( x + y = -3 ).That's a useful equation. Now, can I find another relationship by subtracting the equations? Let me check.First equation minus second equation: ( x^3 + 3x y^2 - 3x^2 y - y^3 = 158 - (-185) = 158 + 185 = 343 )Looking at the left-hand side: ( x^3 - 3x^2 y + 3x y^2 - y^3 ). Wait, that's similar to ( (x - y)^3 ). Let's confirm:( (x - y)^3 = x^3 - 3x^2 y + 3x y^2 - y^3 ). Yes, exactly! So subtracting the second equation from the first gives ( (x - y)^3 = 343 ). Therefore, ( x - y = sqrt[3]{343} = 7 ). So ( x - y = 7 ).Now, we have a system of two linear equations:1. ( x + y = -3 )2. ( x - y = 7 )This seems easier to solve. Let me add these two equations:( (x + y) + (x - y) = -3 + 7 )( 2x = 4 )( x = 2 )Then, substitute back into one of the equations, say ( x + y = -3 ):( 2 + y = -3 )( y = -3 - 2 )( y = -5 )So, the solution is ( x = 2 ), ( y = -5 ). Let me check these values in the original equations to make sure they work.First equation: ( x^3 + 3x y^2 )Plug in x = 2, y = -5:( 2^3 + 3*2*(-5)^2 = 8 + 6*25 = 8 + 150 = 158 ). That's correct.Second equation: ( 3x^2 y + y^3 )Plug in x = 2, y = -5:( 3*(2)^2*(-5) + (-5)^3 = 3*4*(-5) + (-125) = 12*(-5) -125 = -60 -125 = -185 ). That's also correct.Wait, so both equations are satisfied. Therefore, the solution is x = 2, y = -5.But wait, hold on. When we took the cube roots, we assumed real roots, right? Because for real numbers, the cube root of a negative number is negative, and cube root of a positive number is positive. But cube roots can also have complex solutions. However, since the original equations have real coefficients and we found real solutions that satisfy both equations, I think that's the only real solution. But just to be thorough, are there other cube roots we should consider?For the equation ( (x + y)^3 = -27 ), the real cube root is -3, but there are also two complex cube roots. Similarly, ( (x - y)^3 = 343 ) has a real cube root of 7, and two complex roots. But since we're dealing with real numbers here (the original equations are real), the other solutions would involve complex numbers. The problem doesn't specify whether we need complex solutions or just real ones. Given that the problem is presented without complex numbers mentioned, it's safe to assume we're looking for real solutions. Therefore, the only real solution is x = 2, y = -5.But just to be absolutely sure, let me consider if there could be other real solutions. Suppose that when we added and subtracted the equations, we found x + y = -3 and x - y = 7, leading to x = 2, y = -5. But could there be another pair of real numbers x and y that satisfy the original equations without satisfying x + y = -3 and x - y = 7?Let me think. If (x + y)^3 = -27 and (x - y)^3 = 343, then these equations must hold. But in real numbers, the cube function is bijective (one-to-one and onto), so each real number has a unique real cube root. Therefore, the only real solution comes from x + y = -3 and x - y = 7. Thus, no other real solutions exist. So this must be the only solution.Therefore, the answer is x = 2 and y = -5.Final AnswerThe solution is (boxed{x = 2}) and (boxed{y = -5}).</think>Given the system of equations:1. ( x^{3} + 3x y^{2} = 158 )2. ( 3x^{2} y + y^{3} = -185 )We recognize that adding these equations gives:[(x + y)^3 = x^3 + 3x^2 y + 3x y^2 + y^3 = 158 + (-185) = -27]Taking the cube root of both sides, we find:[x + y = -3]Next, subtracting the second equation from the first gives:[(x - y)^3 = x^3 - 3x^2 y + 3x y^2 - y^3 = 158 + 185 = 343]Taking the cube root of both sides, we find:[x - y = 7]We now solve the system of linear equations:1. ( x + y = -3 )2. ( x - y = 7 )Adding these two equations:[2x = 4 implies x = 2]Substituting ( x = 2 ) back into ( x + y = -3 ):[2 + y = -3 implies y = -5]Verification of the solution ( x = 2 ) and ( y = -5 ) in the original equations confirms they are correct. Therefore, the solution is:[boxed{x = 2} quad text{and} quad boxed{y = -5}]]

🔑:Solution.Add and subtract the first and second equations:begin{aligned}& left{begin{array} { l } { x ^ { 3 } + 3 x y ^ { 2 } + 3 x ^ { 2 } y + y ^ { 3 } = - 2 7 , } { x ^ { 3 } + 3 x y ^ { 2 } - 3 x ^ { 2 } y - y ^ { 3 } = 3 4 3 }end{array} Leftrightarrow left{begin{array}{l}(x+y)^{3}=-27, (x-y)^{3}=343end{array} Leftrightarrowright.right. & Leftrightarrowleft{begin{array} { l } { x + y = - 3 , } { x - y = 7 }end{array} Leftrightarrow left{begin{array}{l}x=2 y=-5end{array}right.right.end{aligned}Answer: (2, -5).

🧠:[<think>Okay, so I have this problem here where I need to find the interval where the zero of the function f(x) = ln((3x)/2 - 2/x) is located. The options are A: (0,1), B: (1,2), C: (2,3), D: (3,4). Let me try to figure this out step by step.First, I remember that a zero of a function is where the function crosses the x-axis, which means f(x) = 0. So I need to solve the equation ln((3x)/2 - 2/x) = 0. To solve this, I can exponentiate both sides to get rid of the natural logarithm. That would give me (3x)/2 - 2/x = e^0. Since e^0 is 1, the equation simplifies to (3x)/2 - 2/x = 1. Alright, so now I need to solve (3x)/2 - 2/x = 1 for x. Let me write that out again: (3x)/2 - 2/x = 1. Hmm, this looks like a rational equation. Maybe I can multiply through by 2x to eliminate the denominators. Let's try that. Multiplying each term by 2x gives:2x*(3x)/2 - 2x*(2/x) = 2x*1.Simplifying each term: The first term is (2x)*(3x)/2. The 2 in the numerator and denominator cancels, leaving x*3x = 3x². The second term is 2x*(2/x). The x cancels, so that becomes 2*2 = 4. The right side is 2x*1 = 2x. So putting it all together, the equation becomes:3x² - 4 = 2x.Let me rearrange this into standard quadratic form. Subtract 2x from both sides:3x² - 2x - 4 = 0.Now I need to solve this quadratic equation. Let me check if it factors. The coefficients are 3, -2, -4. Let's see: looking for two numbers that multiply to (3)*(-4) = -12 and add to -2. Hmm, possible factors of -12: -6 and 2. -6 + 2 = -4. Not quite. Maybe -4 and 3? -4 + 3 = -1. Nope. Not obvious. Maybe it doesn't factor nicely. I'll use the quadratic formula. For ax² + bx + c = 0, x = [-b ± sqrt(b² - 4ac)]/(2a). So here, a = 3, b = -2, c = -4.Calculating the discriminant: b² - 4ac = (-2)^2 - 4*3*(-4) = 4 + 48 = 52. That's positive, so two real roots. Then the solutions are:x = [2 ± sqrt(52)]/(2*3) = [2 ± 2*sqrt(13)]/6 = [1 ± sqrt(13)]/3.Wait, sqrt(52) simplifies to 2*sqrt(13), right? Because 52 is 4*13. So sqrt(52) is 2*sqrt(13). Therefore, the solutions are (1 ± sqrt(13))/3.Now, sqrt(13) is approximately 3.6055. So let's compute both roots:First solution: [1 + 3.6055]/3 ≈ 4.6055/3 ≈ 1.535.Second solution: [1 - 3.6055]/3 ≈ (-2.6055)/3 ≈ -0.8685.But since we're dealing with the original function f(x) = ln((3x)/2 - 2/x), we need to consider the domain where the argument of the natural logarithm is positive. So (3x)/2 - 2/x > 0. Also, x cannot be zero because of the 2/x term.So let's check the possible solutions. The second solution is negative, x ≈ -0.8685. Plugging that into (3x)/2 - 2/x: (3*(-0.8685))/2 - 2/(-0.8685) ≈ (-2.6055)/2 + 2.303 ≈ -1.30275 + 2.303 ≈ 1.00025. Which is positive. Wait, but x is negative here. However, the original function has 2/x term. If x is negative, 2/x is negative, and (3x)/2 is also negative. So their difference is (- something) - (- something else). Wait, let me check again.Wait, (3x)/2 - 2/x. If x is negative, then (3x)/2 is negative, and 2/x is negative. So it's (negative) - (negative). Which could be positive or negative. For x ≈ -0.8685, we found that (3x)/2 - 2/x ≈ 1.00025, which is positive. But the problem is, when x is negative, the original function's argument is positive, but x is in the domain of real numbers except zero. So does the original function allow x negative? Let's see.But looking back, the problem is asking for the interval where the zero is located. The options given are all intervals in positive numbers: (0,1), (1,2), etc. So the answer is likely among these. The negative solution is x ≈ -0.8685, but since the options don't include negative intervals, we can disregard that. The positive solution is x ≈ 1.535. So that would fall into interval (1,2), which is option B.But let me verify this. Let me check if the function f(x) is defined in the intervals given. Let's check first the domain of f(x). The argument of the logarithm must be positive: (3x)/2 - 2/x > 0.Let me solve the inequality (3x)/2 - 2/x > 0.Multiply both sides by 2x to eliminate denominators. But I need to be careful with the sign of x when multiplying. Because if x is positive, multiplying doesn't change the inequality direction. If x is negative, it does. But since we have options all in positive intervals, maybe x is positive. Let's assume x > 0 for now.Then, multiplying both sides by 2x (positive) gives:(3x)/2 * 2x - 2/x * 2x > 0 * 2xSimplify:3x² - 4 > 0So 3x² - 4 > 0 → 3x² > 4 → x² > 4/3 → x > sqrt(4/3) or x < -sqrt(4/3). But since x > 0, only x > sqrt(4/3). sqrt(4/3) is approximately 1.1547. So the domain of f(x) is x > sqrt(4/3) ≈ 1.1547.Therefore, the function is only defined for x > approximately 1.1547. Therefore, the zero at x ≈ 1.535 is indeed in the interval (1,2). Therefore, the answer should be B.But let me confirm by plugging in values in the original equation.First, check x=1: (3*1)/2 - 2/1 = 1.5 - 2 = -0.5. The argument of the logarithm is negative, which is invalid. But x=1 is not in the domain, as expected.x=1.5: (3*1.5)/2 - 2/1.5 = 4.5/2 - 1.333... = 2.25 - 1.333 ≈ 0.917. So ln(0.917) is negative. So f(1.5) is negative.x=2: (3*2)/2 - 2/2 = 3 - 1 = 2. ln(2) ≈ 0.693, positive. So f(2) is positive.Wait, but the zero is at x≈1.535. So between 1.5 and 2, the function goes from negative to positive. Therefore, by the Intermediate Value Theorem, since f(x) is continuous on (sqrt(4/3), ∞), which is approximately (1.1547, ∞), and f(1.5) ≈ negative, f(2) ≈ positive, so the zero is in (1.5, 2). But the options given are (1,2). So (1,2) includes this interval, so the correct answer is B: (1,2).Wait, but the domain of f(x) is x > sqrt(4/3) ≈ 1.1547, so x must be greater than approximately 1.1547. Therefore, the interval (1,2) includes the valid domain from 1.1547 onwards. So even though at x=1 the function isn't defined, the interval (1,2) is the correct answer because the zero is at 1.535, which is between 1 and 2.Alternatively, maybe I should check the function at the endpoints of the intervals given.Option B is (1,2). Since the domain starts at ~1.1547, let me check f(1.1547). Let's compute (3x)/2 - 2/x at x = sqrt(4/3). sqrt(4/3) is 2/sqrt(3) ≈ 1.1547. Then:(3*(2/sqrt(3)))/2 - 2/(2/sqrt(3)) = (6/sqrt(3))/2 - (2*sqrt(3))/2 = (3/sqrt(3)) - sqrt(3) = sqrt(3) - sqrt(3) = 0.So at x= sqrt(4/3), the argument is zero. Therefore, as x approaches sqrt(4/3) from the right, the argument approaches zero from the positive side. Therefore, f(x) approaches ln(0+) which tends to -infinity. So at x just above 1.1547, f(x) is very negative. Then, at x=1.5, we saw it's still negative (~ln(0.917)), and at x=2, it's positive (~ln(2)). Therefore, by the Intermediate Value Theorem, there must be a zero between 1.5 and 2, which is within (1,2). Therefore, the answer is B.But just to be thorough, let's check the function at x=1.5 and x=1.6.x=1.5:(3*1.5)/2 - 2/1.5 = 4.5/2 - 1.333... = 2.25 - 1.333 ≈ 0.9167. ln(0.9167) ≈ -0.087. So negative.x=1.6:(3*1.6)/2 - 2/1.6 = 4.8/2 - 1.25 = 2.4 - 1.25 = 1.15. ln(1.15) ≈ 0.1398. Positive.Therefore, the zero is between 1.5 and 1.6. Which is within (1,2). Therefore, answer B is correct.Wait, but just to confirm, perhaps check another point. Let's try x=1.535 as I calculated earlier.At x=1.535:(3*1.535)/2 - 2/1.535 ≈ (4.605)/2 - 1.303 ≈ 2.3025 - 1.303 ≈ 0.9995 ≈ 1. So ln(1) = 0. Therefore, that's the zero. So 1.535 is indeed the solution, which is approximately 1.535, lying in (1,2).So the answer is B: (1,2). So the correct option is B.Final Answerboxed{B}</think>To find the interval where the zero of the function ( f(x) = ln left( frac{3x}{2} - frac{2}{x} right) ) is located, we need to solve ( f(x) = 0 ). This translates to solving:[ln left( frac{3x}{2} - frac{2}{x} right) = 0]Exponentiating both sides, we get:[frac{3x}{2} - frac{2}{x} = e^0 = 1]Multiplying through by ( 2x ) to clear the denominators:[3x^2 - 4 = 2x]Rearranging into standard quadratic form:[3x^2 - 2x - 4 = 0]Using the quadratic formula ( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ) with ( a = 3 ), ( b = -2 ), and ( c = -4 ):[x = frac{2 pm sqrt{4 + 48}}{6} = frac{2 pm sqrt{52}}{6} = frac{1 pm sqrt{13}}{3}]Approximating ( sqrt{13} approx 3.6055 ):- The positive solution is ( x approx frac{1 + 3.6055}{3} approx 1.535 )- The negative solution is disregarded as the options are positive intervals.Next, we check the domain where the argument of the logarithm is positive:[frac{3x}{2} - frac{2}{x} > 0]For ( x > 0 ), solving this inequality gives ( x > sqrt{frac{4}{3}} approx 1.1547 ). The zero ( x approx 1.535 ) lies within the interval ( (1, 2) ).Checking values:- At ( x = 1.5 ), the argument is positive but ( f(1.5) ) is negative.- At ( x = 2 ), ( f(2) ) is positive.By the Intermediate Value Theorem, the zero must be between ( 1.5 ) and ( 2 ), confirming the interval ( (1, 2) ).Thus, the correct answer is (boxed{B}).,<think>Okay, let's try to figure out where the zero of the function f(x) = ln(3x/2 - 2/x) is located. The options are intervals between 0 and 4, so I need to determine which interval contains the x-value that makes f(x) equal to zero.First, I remember that the natural logarithm function ln(y) is zero when y = 1. So, if f(x) = ln(some expression), then the zero of f(x) occurs when that inner expression equals 1. So, I can set up the equation:3x/2 - 2/x = 1That's the key equation to solve. If I can find the x that satisfies this, then I can see which interval it's in. Since it's a multiple-choice question, maybe I don't need to solve it exactly, but just check the intervals given.Let me write down the equation again:(3/2)x - (2/x) = 1Hmm. To solve for x, maybe I can multiply both sides by x to eliminate the denominator. Let's try that:Multiply both sides by x:(3/2)x^2 - 2 = xThen, rearrange the equation to form a quadratic equation:(3/2)x^2 - x - 2 = 0To make it easier, multiply both sides by 2 to eliminate the fraction:3x^2 - 2x - 4 = 0Now, that's a quadratic equation: 3x^2 - 2x - 4 = 0I can use the quadratic formula here. The quadratic formula is x = [-b ± sqrt(b² - 4ac)]/(2a)Here, a = 3, b = -2, c = -4.So, plugging in:x = [2 ± sqrt( (-2)^2 - 4*3*(-4) )]/(2*3)x = [2 ± sqrt(4 + 48)]/6x = [2 ± sqrt(52)]/6sqrt(52) is equal to 2*sqrt(13), so:x = [2 ± 2sqrt(13)]/6 = [1 ± sqrt(13)]/3sqrt(13) is approximately 3.6055, so:x = [1 + 3.6055]/3 ≈ 4.6055/3 ≈ 1.535x = [1 - 3.6055]/3 ≈ -2.6055/3 ≈ -0.8685But since we're dealing with the original function f(x) = ln(3x/2 - 2/x), the argument of the logarithm must be positive. Also, x can't be zero because of the 2/x term. So, x has to be positive, so we discard the negative root. So, the zero is at approximately x ≈ 1.535. So, this is between 1 and 2, right? Because 1.535 is between 1 and 2.Therefore, the answer should be B: (1, 2).But wait, let me check if I did everything correctly. Maybe I made a mistake in the calculations. Let's verify step by step.Starting from f(x) = ln(3x/2 - 2/x). The zero occurs when 3x/2 - 2/x = 1. Let's test x=1.535 in the expression 3x/2 - 2/x.Compute 3*(1.535)/2 = 3*1.535/2 ≈ 4.605/2 ≈ 2.3025Then 2/x = 2/1.535 ≈ 1.302So, 2.3025 - 1.302 ≈ 1.0005, which is approximately 1. So that checks out. Therefore, x ≈1.535 is indeed the solution. Since 1.535 is between 1 and 2, the interval is (1,2), which is option B.But maybe I should check the endpoints of the intervals to confirm using the Intermediate Value Theorem. Let's check the original function f(x) at the endpoints of the intervals given.First, the domain of f(x). The argument of the natural log must be positive:3x/2 - 2/x > 0So, 3x/2 > 2/xMultiply both sides by x (but need to be careful about the sign of x). Since x is in the denominator in 2/x, x cannot be zero, and if x is positive, multiplying both sides by x is okay. So, assuming x > 0:3x^2/2 > 2Multiply both sides by 2: 3x^2 > 4x^2 > 4/3x > sqrt(4/3) ≈ 1.1547So, the domain of f(x) is x > sqrt(4/3) ≈1.1547. Therefore, the function is only defined for x > approximately 1.1547. So, in the given intervals, A is (0,1), which is partly in the domain (only from ~1.1547 to 1), but mostly not. But the zero is at ~1.535, so x must be in the domain, so the zero is in (1,2). Let me check f(1) and f(2):Wait, but x=1 is in the domain? Let's check if x=1 is in the domain. At x=1:3*(1)/2 - 2/1 = 1.5 - 2 = -0.5. Which is negative, so the argument is negative. Therefore, x=1 is not in the domain. Similarly, x=1.1547 is the lower limit.So, the function is defined for x > sqrt(4/3) ≈1.1547. Therefore, when testing the intervals, the left end of the interval has to be greater than ~1.1547. The intervals given are:A: (0,1) – mostly not in the domain except from ~1.1547 to 1, but the zero is at 1.535, which is not in this interval.B: (1,2) – since 1.1547 is approximately 1.1547, so (1.1547, 2) is the domain. The zero at ~1.535 is in here.C: (2,3) – the zero is at ~1.535, which is less than 2, so not here.D: (3,4) – same reasoning.Therefore, the answer is B.But maybe to confirm, use the Intermediate Value Theorem. Let's compute f(x) at x=1.5 and x=2.First, check at x=1.5 (which is in (1,2)):3*(1.5)/2 - 2/(1.5) = (4.5)/2 - (1.333...) = 2.25 - 1.333... ≈0.9167Then f(1.5) = ln(0.9167) ≈ -0.087, which is negative.Then at x=2:3*2/2 - 2/2 = 3 -1 = 2f(2)=ln(2) ≈0.693, which is positive.So, f(1.5) is negative, f(2) is positive. Since f is continuous on its domain (assuming it's continuous where the argument is positive), then by Intermediate Value Theorem, there is a zero between 1.5 and 2. Wait, but our previous calculation said the zero is at ~1.535. Wait, but 1.535 is between 1.5 and 2. So, maybe better to check at x=1.5 and x=2. Wait, but the options are (1,2). So, f(x) is negative at x=1.5 and positive at x=2, which means the zero is between 1.5 and 2, which is within (1,2). So, that still falls under option B.But also, earlier when I calculated the exact solution, it was approximately 1.535, which is between 1 and 2. Therefore, the answer is B.Wait, but perhaps I should check at x=1. Let's see:Wait, x=1 is not in the domain because 3*1/2 -2/1=1.5 -2=-0.5, which is negative. So, x=1 is not in the domain. Then at x=1.1547, the argument is zero. Wait, but sqrt(4/3)=2/sqrt(3)≈1.1547. At x=2/sqrt(3), let's compute 3x/2 -2/x:3*(2/sqrt(3))/2 -2/(2/sqrt(3)) = (3*2)/(2*sqrt(3)) - (2*sqrt(3))/2 = 3/sqrt(3) - sqrt(3) = sqrt(3) - sqrt(3) =0.Therefore, the domain is x>2/sqrt(3)≈1.1547. So, the interval (1,2) includes x=1.1547 to 2, which is part of the domain.But the zero is at x≈1.535, which is in (1,2). So, yes, the answer is B.Alternatively, maybe check f(1.2) and f(1.5) to see the sign change. Let's compute f(1.2):First, check if x=1.2 is in the domain. Compute 3*(1.2)/2 -2/(1.2):3*1.2 =3.6, divided by 2 is 1.82/1.2≈1.6667So, 1.8 -1.6667≈0.1333Therefore, f(1.2)=ln(0.1333)≈-2.015, which is negative.Wait, but earlier at x=1.5, f(x)=ln(0.9167)≈-0.087, which is also negative. Wait, but at x=2, f(x)=ln(2)=positive. Wait, so from x=1.5 to x=2, the function goes from negative to positive, meaning the zero is between 1.5 and 2. So, in that case, within the interval (1,2), the zero is between 1.5 and 2. But the options are (0,1), (1,2), (2,3), (3,4). So, even though the zero is between 1.5 and 2, that's still within (1,2). So, B is the answer.Alternatively, maybe check f(1.5) and f(1.6):At x=1.5, as before, f(x)=ln(0.9167)≈-0.087.At x=1.6:3*1.6/2 -2/1.6= (4.8)/2 -1.25=2.4 -1.25=1.15f(1.6)=ln(1.15)≈0.1398, which is positive. So, between 1.5 and 1.6, the function crosses zero. So, the zero is in (1.5,1.6), which is within (1,2). So, the correct interval is B.Therefore, I think the answer is B.Final Answerboxed{B}</think>To determine the interval where the zero of the function ( f(x) = ln left( frac{3x}{2} - frac{2}{x} right) ) is located, we need to find when the argument of the logarithm equals 1. This is because the natural logarithm function (ln(y)) is zero when ( y = 1 ).Set up the equation:[frac{3x}{2} - frac{2}{x} = 1]Multiply both sides by ( x ) to eliminate the denominator:[frac{3x^2}{2} - 2 = x]Rearrange to form a quadratic equation:[3x^2 - 2x - 4 = 0]Solve using the quadratic formula ( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ) where ( a = 3 ), ( b = -2 ), and ( c = -4 ):[x = frac{2 pm sqrt{4 + 48}}{6} = frac{2 pm sqrt{52}}{6} = frac{1 pm sqrt{13}}{3}]Approximating ( sqrt{13} approx 3.6055 ):[x approx frac{1 + 3.6055}{3} approx 1.535 quad text{(positive root)}]The positive root ( x approx 1.535 ) is between 1 and 2. To confirm, check the function values at the endpoints of the interval (1, 2):- At ( x = 1.5 ), the argument is approximately 0.9167, so ( f(1.5) approx ln(0.9167) approx -0.087 ) (negative).- At ( x = 2 ), the argument is 2, so ( f(2) = ln(2) approx 0.693 ) (positive).By the Intermediate Value Theorem, the function changes sign between 1.5 and 2, confirming a zero in the interval (1, 2).Thus, the interval where the zero of the function is located is (boxed{B}).]

🔑:Solution: The function f(x) = ln left( frac{3x}{2} - frac{2}{x} right) is increasing on the interval (0, +infty), Also, since f(1) = ln left( frac{3}{2} right) - 2 0, Therefore, the function f(x) has one zero in the interval (1, 2), Hence, the correct choice is: boxed{text{B}} First, analyze the monotonicity of the function, and then combine the theorem of the existence of zeros, it can be concluded that the function f(x) has one zero in the interval (1, 2). This question examines the theorem of determining the zeros of a function. Mastering the method and steps to judge the position of the zeros of a function is the key to solving the problem.

🧠:[<think>Okay, let's try to solve this problem step by step. The problem says that we have two complex numbers, z₁ and z₂, which are symmetric about the line y = x in the complex plane. We know that z₁ is 1 + 2i, and we need to find the value of z₁/z₂ from the given options.First, I need to recall what it means for two points to be symmetric about the line y = x. Symmetry about the line y = x means that if you reflect one point over the line y = x, you get the other point. In other words, if z₁ corresponds to the point (a, b) in the complex plane, then z₂ should correspond to the point (b, a). So, reflecting over y = x swaps the x and y coordinates.Given that z₁ is 1 + 2i, which corresponds to the point (1, 2), then z₂, being symmetric about y = x, should correspond to the point (2, 1). Therefore, z₂ should be 2 + 1i, or simply 2 + i.So, z₁ is 1 + 2i and z₂ is 2 + i. Now, we need to compute z₁/z₂.To divide complex numbers, I remember that we can multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. Multiplying by the conjugate helps eliminate the imaginary part in the denominator.So, let's write that out:z₁ / z₂ = (1 + 2i) / (2 + i)Multiply numerator and denominator by the conjugate of the denominator, which is (2 - i):= [(1 + 2i)(2 - i)] / [(2 + i)(2 - i)]First, let's compute the denominator:(2 + i)(2 - i) = 2*2 + 2*(-i) + i*2 + i*(-i) = 4 - 2i + 2i - i²Simplify:The -2i and +2i cancel out, so we have 4 - i². Since i² = -1, this becomes 4 - (-1) = 4 + 1 = 5. So the denominator is 5.Now the numerator:(1 + 2i)(2 - i) = 1*2 + 1*(-i) + 2i*2 + 2i*(-i)Compute each term:1*2 = 21*(-i) = -i2i*2 = 4i2i*(-i) = -2i²Combine the terms:2 - i + 4i - 2i²Combine like terms:2 + ( -i + 4i ) + (-2i²) = 2 + 3i - 2i²Again, since i² = -1, this becomes:2 + 3i - 2*(-1) = 2 + 3i + 2 = 4 + 3iSo the numerator is 4 + 3i, and the denominator is 5.Therefore, z₁/z₂ = (4 + 3i)/5 = 4/5 + 3/5 iLooking at the answer options:A: 3/5 - 4/5iB: 3/5 + 4/5iC: 4/5 - 3/5iD: 4/5 + 3/5iSo, the answer should be D: 4/5 + 3/5i.Wait a minute, let me double-check. Let's confirm if my reflection was correct. The original point is (1,2). Reflecting over y = x should be (2,1), which is z₂ = 2 + i. Then, z₁/z₂ is (1 + 2i)/(2 + i). Did I compute that correctly?Yes, I multiplied numerator and denominator by (2 - i), which gives numerator (1 + 2i)(2 - i) and denominator (2 + i)(2 - i) = 5. Then expanding the numerator:1*2 = 21*(-i) = -i2i*2 = 4i2i*(-i) = -2i² = +2So total numerator: 2 - i + 4i + 2 = (2 + 2) + ( -i + 4i ) = 4 + 3i. So yes, 4 + 3i over 5 is 4/5 + 3/5i, which is option D. So I think that's correct.Alternatively, another way to check is to consider that reflection over y = x is equivalent to taking the complex conjugate and then swapping real and imaginary parts. Wait, is there a formula for reflecting a complex number over y = x?Let me think. If z = a + bi, then reflection over y = x is b + ai. So, z₂ = (Im(z₁) + Re(z₁)i) = 2 + 1i, which matches. So z₂ is indeed 2 + i.Therefore, the division is correct. So the answer is D.Final Answerboxed{D}</think>Given complex numbers ( z_1 ) and ( z_2 ) which are symmetric about the line ( y = x ) in the complex plane, and ( z_1 = 1 + 2i ), we need to find the value of ( frac{z_1}{z_2} ).1. Reflection about the line ( y = x ): - The complex number ( z_1 = 1 + 2i ) corresponds to the point ( (1, 2) ) in the complex plane. - Reflecting ( (1, 2) ) over the line ( y = x ) gives the point ( (2, 1) ), which corresponds to the complex number ( z_2 = 2 + i ).2. Division of complex numbers: - We need to compute ( frac{z_1}{z_2} = frac{1 + 2i}{2 + i} ). - Multiply the numerator and the denominator by the conjugate of the denominator ( (2 - i) ): [ frac{1 + 2i}{2 + i} cdot frac{2 - i}{2 - i} = frac{(1 + 2i)(2 - i)}{(2 + i)(2 - i)} ]3. Simplify the denominator: [ (2 + i)(2 - i) = 2^2 - (i)^2 = 4 - (-1) = 5 ]4. Simplify the numerator: [ (1 + 2i)(2 - i) = 1 cdot 2 + 1 cdot (-i) + 2i cdot 2 + 2i cdot (-i) ] [ = 2 - i + 4i - 2i^2 = 2 + 3i - 2(-1) = 2 + 3i + 2 = 4 + 3i ]5. Result: [ frac{4 + 3i}{5} = frac{4}{5} + frac{3}{5}i ]Thus, the value of ( frac{z_1}{z_2} ) is (boxed{D}).,<think>Okay, let's see. The problem says that we have two complex numbers, z₁ and z₂, which are symmetric about the line y = x in the complex plane. Given that z₁ is 1 + 2i, we need to find z₁ divided by z₂ and choose the correct answer from the options provided.First, let me recall what symmetry about the line y = x means. In the coordinate plane, reflecting a point over the line y = x swaps its x and y coordinates. So, if we have a point (a, b), its reflection over y = x would be (b, a). Translating this into complex numbers, if z₁ corresponds to the point (a, b), then z₂ should correspond to (b, a). Therefore, if z₁ is a + bi, then z₂ would be b + ai. Let me verify that. For example, if z₁ is 1 + 2i, which corresponds to the point (1, 2), then reflecting over y = x would give the point (2, 1), which corresponds to the complex number 2 + 1i, or 2 + i. So, z₂ should be 2 + i. Wait, but the options given are fractions. Let me check if I did that correctly. Let me confirm: if z₁ is 1 + 2i, then its reflection over y = x is indeed 2 + i. But when I look at the answer choices, they are fractions like 3/5 - 4/5i. Hmm, so maybe there's more to this. Maybe z₂ isn't simply swapping the real and imaginary parts? Or perhaps I need to compute the ratio z₁/z₂ and see which of those options it matches?Wait, let's try that. If z₂ is 2 + i, then compute z₁/z₂. So, z₁ is 1 + 2i and z₂ is 2 + i. To divide them, we can multiply numerator and denominator by the conjugate of the denominator. So:(1 + 2i)/(2 + i) = [(1 + 2i)(2 - i)] / [(2 + i)(2 - i)].First compute the numerator: (1)(2) + (1)(-i) + (2i)(2) + (2i)(-i) = 2 - i + 4i - 2i².Since i² is -1, so -2i² is +2. So, combining terms: 2 + 3i + 2 = 4 + 3i.Denominator: (2 + i)(2 - i) = 2² - (i)² = 4 - (-1) = 5.Therefore, (1 + 2i)/(2 + i) = (4 + 3i)/5 = 4/5 + 3/5i, which is option D. But option D is 4/5 + 3/5i, which is one of the choices. But wait, the options are A, B, C, D. So if the answer is D, then that would be the choice. However, let me check if my assumption that z₂ is simply 2 + i is correct.Wait, maybe the symmetry over y = x in the complex plane is a bit more involved. Let me think. The line y = x in the complex plane is the set of points where the real part equals the imaginary part. Reflecting a complex number over this line would indeed swap the real and imaginary parts. So, if z₁ = a + bi, then its reflection z₂ would be b + ai. So for z₁ = 1 + 2i, z₂ should be 2 + i. So that seems correct. Then the ratio z₁/z₂ would be (1 + 2i)/(2 + i) = 4/5 + 3/5i, which is option D. So why is this not one of the other options?Wait, maybe I made a mistake in the calculation. Let me double-check. Let me do the multiplication again. The numerator: (1 + 2i)(2 - i). Let's expand this:1*2 = 21*(-i) = -i2i*2 = 4i2i*(-i) = -2i² = -2*(-1) = 2So adding all these terms: 2 - i + 4i + 2 = (2 + 2) + (-i + 4i) = 4 + 3i. So numerator is 4 + 3i. Denominator is 5. So yes, (4 + 3i)/5 is 4/5 + 3/5i, which is D. So the answer should be D. But wait, the options given are A: 3/5 - 4/5i, B: 3/5 + 4/5i, C: 4/5 - 3/5i, D:4/5 + 3/5i. So D is the correct one. But let me check again if z₂ is indeed 2 + i.Alternatively, perhaps the reflection over y=x is not just swapping real and imaginary parts. Wait, in the complex plane, the reflection over the line y=x can be represented by the transformation z ↦ (conjugate(z)) rotated by 90 degrees or something? Wait, perhaps not. Let me think.In the standard complex plane, the real axis is horizontal, and the imaginary axis is vertical. So, the line y = x is a 45-degree line. Reflecting a point over y = x swaps the x and y coordinates. So, if you have a complex number z = a + bi, which corresponds to the point (a, b), then reflecting over y = x would give the point (b, a), which corresponds to the complex number b + ai. So yes, z₂ is b + ai. So for z₁ = 1 + 2i, z₂ = 2 + i. Then, z₁/z₂ is (1 + 2i)/(2 + i) = 4/5 + 3/5i, which is D. So that should be the answer.But let me check if there's another approach. Maybe the problem is considering the line y = x in the complex plane, so in terms of complex numbers, that line can be represented as the set { z ∈ ℂ | Re(z) = Im(z) }. A reflection over this line can be represented as a linear transformation. To find the reflection of a complex number over the line y = x, perhaps we can use some formula. Let me recall that in general, reflecting a point over a line in the plane can be done with some transformation matrix.In linear algebra, the reflection over the line y = x is represented by the matrix [[0, 1], [1, 0]]. So, if we have a point (a, b), the reflection is (b, a). So, in complex numbers, that would be turning a + bi into b + ai. So, again, z₂ = 2 + i if z₁ is 1 + 2i. So, that seems consistent.Alternatively, maybe we can think of this in terms of complex conjugation. Let me recall that complex conjugation, which reflects over the real axis, is equivalent to changing the sign of the imaginary part. But reflecting over y = x is a different operation. Alternatively, if we let T be the reflection over y = x, then for a complex number z = x + yi, T(z) = y + xi. So, T(z) = Im(z) + Re(z)i. Which is exactly swapping the real and imaginary parts. Therefore, for z₁ = 1 + 2i, z₂ = 2 + i. Then, z₁/z₂ = (1 + 2i)/(2 + i) = (1 + 2i)(2 - i)/ ( (2 + i)(2 - i) ) = (2 - i + 4i - 2i²)/ (4 + 1) ) = (2 + 3i + 2)/5 = (4 + 3i)/5 = 4/5 + 3/5i, which is option D. Therefore, the answer should be D. But let me check why the options have other fractions. Maybe I did something wrong here. Alternatively, perhaps the question is referring to symmetry about the line y = x, but not the reflection. Wait, symmetry about a line means that the line is the perpendicular bisector of the segment joining the two points. So, reflecting one point over the line gives the other point, which is what I considered. So, if z₁ is 1 + 2i, then z₂ is 2 + i. Therefore, the ratio z₁/z₂ is (1 + 2i)/(2 + i) = 4/5 + 3/5i, which is D.But let me check with another method. Let me write z₂ as the reflection of z₁ over y = x. If z₁ is (1,2), then z₂ is (2,1), which is 2 + i. Then, compute z₁/z₂:(1 + 2i)/(2 + i). Multiply numerator and denominator by the conjugate of the denominator:(1 + 2i)(2 - i) / (2 + i)(2 - i) = (2 - i + 4i - 2i²)/ (4 + 1) = (2 + 3i + 2)/5 = (4 + 3i)/5.Yes, that's 4/5 + 3/5i. So, D is the answer. Therefore, the correct option is D.But just to be thorough, let me make sure that there isn't a different interpretation of symmetry about the line y = x. For instance, sometimes people might refer to symmetry as in the points are mirrored over the line, but maybe there's a scaling factor involved? But in standard geometric terms, symmetry about a line means reflection over that line, which swaps the coordinates as we did. Alternatively, could the problem be referring to the conjugate? For example, if you take the complex conjugate of z₁, that reflects over the real axis. But reflecting over y = x is different. Alternatively, maybe the mapping for reflection over y = x in complex numbers can be written as z ↦ i * conjugate(z), let's check. Let z = x + yi. Then, conjugate(z) = x - yi. Multiplying by i gives i*(x - yi) = xi - y i² = xi + y. Which is y + xi, which is exactly the reflection over y = x. So, yes, reflecting over y = x can be represented by z ↦ i * conjugate(z). So, z₂ = i * conjugate(z₁). Let's compute that. Given z₁ = 1 + 2i, conjugate(z₁) = 1 - 2i. Then, multiply by i: i*(1 - 2i) = i - 2i² = i + 2. So, z₂ = 2 + i, which matches our earlier result. Therefore, z₂ = 2 + i. Therefore, the ratio z₁/z₂ is (1 + 2i)/(2 + i) = (4 + 3i)/5, which is 4/5 + 3/5i, option D. So that seems correct. Alternatively, let me check if the problem could be referring to some other symmetry. For example, if the points are symmetric about the line y = x, but not necessarily reflections. But in standard terminology, symmetry about a line for two points means that the line is the perpendicular bisector of the segment connecting them, which is equivalent to one point being the reflection of the other over the line. Therefore, z₂ must be the reflection of z₁ over y = x, which gives z₂ = 2 + i. Alternatively, let's suppose there is a miscalculation here. Let me verify with another approach. Suppose we represent the complex numbers as vectors. The reflection of a vector over the line y = x would swap the x and y components. So, the vector from the origin to z₁ is (1,2), so reflecting over y = x gives (2,1), which is z₂ = 2 + i. Therefore, the ratio is (1 + 2i)/(2 + i). Calculating again:Multiply numerator and denominator by the conjugate of the denominator:(1 + 2i)(2 - i) / (2 + i)(2 - i)Compute numerator:1*2 = 21*(-i) = -i2i*2 = 4i2i*(-i) = -2i² = 2Add them up: 2 - i + 4i + 2 = 4 + 3iDenominator: 4 - (i)^2 = 4 - (-1) = 5Therefore, the result is (4 + 3i)/5 = 4/5 + 3/5i, which is option D. So, the answer must be D.Wait, but let me check if any of the other options could result from a different interpretation. For example, if instead of swapping real and imaginary parts, the reflection was over the line y = x but in some other way. But no, reflection over y = x in standard geometry is precisely swapping the coordinates. Alternatively, suppose someone thought that reflecting over y = x involves rotating the plane by 45 degrees, reflecting over the x-axis, then rotating back. But that seems more complicated. Let me see. The reflection over y = x can also be represented as a rotation. For example, rotating the plane by -45 degrees, reflecting over the x-axis, and then rotating back by 45 degrees. But that might complicate things.Alternatively, let's consider the formula for reflection over a line in the complex plane. The general formula for reflecting a complex number z over a line can be given by:If the line is represented in the form e^{iθ}(real axis), then the reflection of z over this line is e^{2iθ} * conjugate(z). In our case, the line y = x makes a 45-degree angle with the real axis. So θ = 45 degrees, or π/4 radians. Then, e^{2iθ} = e^{iπ/2} = i. Therefore, the reflection of z over the line y = x is i * conjugate(z). Which matches what I had earlier. So, z₂ = i * conjugate(z₁). Let me compute this again for z₁ = 1 + 2i. conjugate(z₁) = 1 - 2i. Multiply by i: i*(1 - 2i) = i - 2i² = i + 2. Therefore, z₂ = 2 + i. Same result. Then, z₁/z₂ = (1 + 2i)/(2 + i) = (4 + 3i)/5. So D.Therefore, all approaches lead to the same conclusion. So, the answer is D. Therefore, I think the correct option is D.Final Answerboxed{D}</think>Given complex numbers ( z_1 ) and ( z_2 ) whose corresponding points in the complex plane are symmetric about the line ( y = x ), and ( z_1 = 1 + 2i ), we need to find ( frac{z_1}{z_2} ).1. Symmetry about the line ( y = x ): - Reflecting a point ( (a, b) ) over the line ( y = x ) results in the point ( (b, a) ). - For the complex number ( z_1 = 1 + 2i ) (corresponding to the point ( (1, 2) )), its reflection over ( y = x ) is ( z_2 = 2 + i ) (corresponding to the point ( (2, 1) )).2. Calculating ( frac{z_1}{z_2} ): - ( z_1 = 1 + 2i ) - ( z_2 = 2 + i ) - Multiply the numerator and denominator by the conjugate of the denominator: [ frac{1 + 2i}{2 + i} = frac{(1 + 2i)(2 - i)}{(2 + i)(2 - i)} ] - Expand the numerator: [ (1 + 2i)(2 - i) = 1 cdot 2 + 1 cdot (-i) + 2i cdot 2 + 2i cdot (-i) = 2 - i + 4i - 2i^2 ] [ = 2 + 3i + 2 = 4 + 3i quad (text{since } i^2 = -1) ] - Expand the denominator: [ (2 + i)(2 - i) = 2^2 - i^2 = 4 - (-1) = 5 ] - Combine the results: [ frac{4 + 3i}{5} = frac{4}{5} + frac{3}{5}i ]Thus, the value of ( frac{z_1}{z_2} ) is ( frac{4}{5} + frac{3}{5}i ), which corresponds to option D.[boxed{D}]]

🔑:Since the complex numbers z_1 and z_2 correspond to points in the complex plane that are symmetric about the line y=x, and z_1=1+2i, it follows that z_2=2+i. Therefore, frac {z_{1}}{z_{2}} equals frac {1+2i}{2+i}, which simplifies to frac {(1+2i)(2-i)}{(2+i)(2-i)}, resulting in frac {4}{5}+ frac {3}{5}i. Thus, the correct choice is: boxed{text{D}}. Given z_1=1+2i, we can deduce z_2=2+i. By applying the rules of complex number operations, we can arrive at the solution. This problem tests the understanding of symmetry about the line y=x, the operations with complex numbers, and assesses reasoning and computational skills. It is considered a basic question.

🧠:[<think>Okay, so I have this problem here: a right triangle with a perimeter of 4 plus the square root of 26, and the median to the hypotenuse is 2. I need to find the area of this triangle. Hmm, let's start by recalling some properties of right triangles.First off, in a right triangle, the hypotenuse is the longest side, right? And the median to the hypotenuse is half the hypotenuse. Wait, that seems important. Let me verify that. So, the median from the right angle to the hypotenuse—actually, in any triangle, the median to a side is the line from the midpoint of that side to the opposite vertex. But in a right triangle, the midpoint of the hypotenuse is equidistant from all three vertices. So, the median to the hypotenuse is half the hypotenuse. Therefore, if the median is 2, then the hypotenuse must be 4. Because 2 times 2 is 4. So that's a key point. Let me write that down.Hypotenuse (c) = 2 * median = 2 * 2 = 4.Okay, so the hypotenuse is 4. Now, the perimeter is given as 4 + sqrt(26). The perimeter of a triangle is the sum of its sides. So, if the hypotenuse is 4, then the other two sides (let's call them a and b) must add up to (4 + sqrt(26)) minus 4, which is sqrt(26). So, a + b = sqrt(26). Got that.Also, since it's a right triangle, by the Pythagorean theorem, a² + b² = c² = 16.So, now we have two equations:1. a + b = sqrt(26)2. a² + b² = 16Hmm, I need to find the area, which is (a*b)/2. So, if I can find the product of a and b, then divide by 2, I can get the area.I remember that (a + b)² = a² + 2ab + b². So, maybe I can use that to find ab.Let's compute (a + b)²:(a + b)² = (sqrt(26))² = 26.But also, (a + b)² = a² + 2ab + b². From the Pythagorean theorem, a² + b² = 16, so substitute that in:26 = 16 + 2ab.Subtract 16 from both sides:26 - 16 = 2ab => 10 = 2ab => ab = 5.Therefore, the area is (a*b)/2 = 5/2 = 2.5. But let me check that again. Wait, ab = 5, so 5 divided by 2 is indeed 2.5, which is 5/2. So, the area would be 5/2. Is that right? Let me verify.Wait, perimeter is 4 + sqrt(26). The hypotenuse is 4, so the other two sides sum to sqrt(26). Their squares sum to 16. Then (sqrt(26))² is 26, so 26 = 16 + 2ab, which gives ab = 5. So area is 5/2. Hmm, that seems straightforward. But let me check if I made any wrong assumptions.First, median to hypotenuse is 2, so hypotenuse is 4. That seems correct because in a right triangle, the median to the hypotenuse is half the hypotenuse. That's a standard result. Let me confirm: in a right triangle, the median to the hypotenuse is equal to half the hypotenuse. Yes, because the midpoint of the hypotenuse is the circumradius, which is half the hypotenuse. So that's correct.Then, perimeter is 4 + sqrt(26). If the hypotenuse is 4, then the sum of the legs is sqrt(26). Then, using (a + b)^2 = a² + 2ab + b². We know a² + b² = 16, so substituting gives us 26 = 16 + 2ab. Then 2ab = 10, so ab = 5, so area is 5/2. That seems to check out.But wait a second, maybe I should check if such a triangle actually exists. Because just because we have a + b = sqrt(26) and ab = 5, maybe the legs are real numbers. Let's see.Let’s consider a and b as roots of the quadratic equation x² - (a + b)x + ab = 0. So, x² - sqrt(26)x + 5 = 0. Let's compute the discriminant:Discriminant D = [sqrt(26)]² - 4*1*5 = 26 - 20 = 6. Since D is positive, there are two real roots. Therefore, such a triangle exists. So, that's good.Alternatively, I can compute the sides. Let me solve for a and b.From a + b = sqrt(26) and a² + b² = 16.Wait, but if a + b = sqrt(26), then (sqrt(26))² = a² + 2ab + b² = 26. But a² + b² is 16, so 16 + 2ab = 26. Therefore, 2ab = 10, ab = 5. So, the area is 5/2. Therefore, the answer is 5/2. So, that's 2.5. So, boxed, that would be boxed{dfrac{5}{2}}.Wait, but let me just cross-verify once again. Let's suppose the legs are a and b, hypotenuse c=4. Perimeter a + b + c = 4 + sqrt(26). So, a + b = sqrt(26). Then, a² + b² = 16. Then, (a + b)^2 = 26 = a² + 2ab + b² = 16 + 2ab. So, 2ab = 10, ab = 5. Therefore, area is 5/2. That seems consistent.Alternatively, if I pick actual numbers. Suppose a and b are such that a + b = sqrt(26), and ab = 5. Let's find a and b. The quadratic equation is x² - sqrt(26)x + 5 = 0. Then, solutions are [sqrt(26) ± sqrt(26 - 20)] / 2 = [sqrt(26) ± sqrt(6)] / 2. So, the legs would be [sqrt(26) + sqrt(6)] / 2 and [sqrt(26) - sqrt(6)] / 2. Let's check if their squares add to 16.Compute ([sqrt(26)+sqrt(6)]/2)^2 + ([sqrt(26)-sqrt(6)]/2)^2.First term: (26 + 2*sqrt(156) + 6)/4 = (32 + 2*sqrt(156))/4.Second term: (26 - 2*sqrt(156) + 6)/4 = (32 - 2*sqrt(156))/4.Add them: (32 + 2*sqrt(156) + 32 - 2*sqrt(156))/4 = 64/4 = 16. Perfect, that matches the Pythagorean theorem. So, all the numbers check out. Therefore, the area is indeed 5/2.Therefore, I think the answer is 5/2.Final AnswerThe area of the triangle is boxed{dfrac{5}{2}}.</think>Given a right triangle with a perimeter of (4 + sqrt{26}) and a median length of 2 on the hypotenuse, we need to find the area of this triangle.1. Hypotenuse Calculation: - The median to the hypotenuse in a right triangle is half the hypotenuse. Therefore, the hypotenuse (c) is (2 times 2 = 4).2. Perimeter Information: - The perimeter of the triangle is (4 + sqrt{26}). Since the hypotenuse is 4, the sum of the other two sides (a) and (b) is ((4 + sqrt{26}) - 4 = sqrt{26}).3. Using Pythagorean Theorem: - We know (a + b = sqrt{26}) and (a^2 + b^2 = 16) (since (c = 4)).4. Finding the Product (ab): - Using the identity ((a + b)^2 = a^2 + 2ab + b^2): [ (sqrt{26})^2 = 16 + 2ab ] [ 26 = 16 + 2ab ] [ 10 = 2ab implies ab = 5 ]5. Calculating the Area: - The area of the triangle is (frac{1}{2}ab): [ text{Area} = frac{1}{2} times 5 = frac{5}{2} ]Thus, the area of the triangle is (boxed{dfrac{5}{2}}).,<think>Okay, let me try to solve this problem step by step. So, we have a right triangle with a perimeter of 4 + sqrt(26), and the median to the hypotenuse is 2. We need to find the area of this triangle. Hmm, right triangles... medians... perimeters. Let me recall some properties.First, in a right triangle, the median to the hypotenuse is half the length of the hypotenuse. Wait, but the problem says the median length is 2. So, if the median is half the hypotenuse, then the hypotenuse should be 4. But let me confirm that. Yeah, in a right triangle, the hypotenuse's median is indeed half the hypotenuse. Because the median connects the right angle to the midpoint of the hypotenuse, forming two isosceles triangles. So, that would make the hypotenuse 4. So, hypotenuse c = 4. Got that.Now, the perimeter is given as 4 + sqrt(26). The perimeter of a triangle is the sum of its sides. So, if the hypotenuse is 4, then the sum of the other two sides (let's call them a and b) should be (4 + sqrt(26)) - 4 = sqrt(26). So, a + b = sqrt(26). Okay, that's one equation.We also know that in a right triangle, a^2 + b^2 = c^2. Since c is 4, that's a^2 + b^2 = 16. So, we have two equations:1. a + b = sqrt(26)2. a^2 + b^2 = 16Hmm, I need to find the area, which is (a*b)/2. So, if I can find the product ab, then I can compute the area. Let me recall that (a + b)^2 = a^2 + 2ab + b^2. So, we can use this to find ab.From equation 1, (a + b)^2 = (sqrt(26))^2 = 26. From equation 2, a^2 + b^2 = 16. Therefore, substituting into the first expansion: 26 = 16 + 2ab. So, 2ab = 26 - 16 = 10. Therefore, ab = 5. So, the area is 5/2, which is 2.5. Wait, but the answer is supposed to be in a box and probably as a fraction. So, 5/2. But let me check if I did everything correctly.Wait, but hold on. The problem states that the median to the hypotenuse is 2. But in a right triangle, the median to the hypotenuse is always half the hypotenuse. So, if the median is 2, then the hypotenuse is 4. That seems right. Then, perimeter is 4 + sqrt(26). So, the sum of a and b is sqrt(26). Then, we have a^2 + b^2 = 16. Then, (a + b)^2 = 26 = 16 + 2ab => 2ab = 10 => ab = 5. Therefore, area is 5/2, which is 2.5. So, the area is 5/2. That seems straightforward. But wait, let me check if the problem is stated correctly.Wait, the perimeter is 4 + sqrt(26). The hypotenuse is 4, so the other two sides add up to sqrt(26). Then, using Pythagoras, their squares add to 16. So, (sqrt(26))^2 is 26, which equals a^2 + 2ab + b^2. Then, since a^2 + b^2 = 16, 16 + 2ab = 26 => 2ab = 10 => ab = 5. Therefore, area is 5/2. So, 5/2. Hmm. That seems correct. But is there a possible mistake here?Wait, let me verify. Suppose we take a right triangle with hypotenuse 4. Let me choose some a and b such that a + b = sqrt(26) and a^2 + b^2 = 16. Let me compute (a + b)^2 = 26, which equals a^2 + 2ab + b^2. Then, since a^2 + b^2 = 16, 26 = 16 + 2ab => 2ab = 10 => ab = 5. So, the area is 5/2, which is 2.5. So, seems okay.Wait, but let me test with actual numbers. Suppose a and b are such that a + b = sqrt(26) ≈ 5.1, and a^2 + b^2 = 16. So, if a and b are both around 2.55, then their squares would be around 6.5 each, which sums to 13, but we need 16. Hmm, that's inconsistent. Wait, that suggests that the product is 5, but a and b each squared sum to 16. Wait, if a and b are real numbers such that a + b = sqrt(26) and a^2 + b^2 = 16, then solving these equations would give ab = 5. But does such a triangle exist?Wait, let's check discriminant. Let me consider a and b as roots of the quadratic equation x^2 - (a + b)x + ab = 0. So, x^2 - sqrt(26)x + 5 = 0. The discriminant would be (sqrt(26))^2 - 4*1*5 = 26 - 20 = 6. So, discriminant is positive, so real roots exist. So, there exist real numbers a and b. Therefore, such a triangle exists. Therefore, the area is 5/2. Hmm. So, the answer is 5/2. So, boxed, that's boxed{dfrac{5}{2}}.But wait, let me double-check my reasoning. The key point here is that in a right triangle, the median to the hypotenuse is half the hypotenuse. So, if the median is 2, hypotenuse is 4. Then, the perimeter is 4 + sqrt(26), so a + b = sqrt(26). Then, by Pythagoras, a^2 + b^2 = 16. Then, (a + b)^2 = 26 = a^2 + b^2 + 2ab = 16 + 2ab => 2ab = 10 => ab = 5. Therefore, area is 5/2. That seems right.Alternatively, maybe there's a different interpretation. Wait, the problem says "a median length of 2 on the hypotenuse". Does that mean the median to the hypotenuse is 2, which we have considered, or maybe a different median? Wait, in a triangle, a median is a line from a vertex to the midpoint of the opposite side. So, the median to the hypotenuse is the one we considered, which in a right triangle is half the hypotenuse. So, if hypotenuse is c, then the median is c/2. So, median is 2, so c = 4. So, that's correct. So, hypotenuse is 4. Then, perimeter is 4 + sqrt(26), so a + b = sqrt(26). Then, Pythagoras gives a^2 + b^2 = 16. Then, (a + b)^2 = 26 = 16 + 2ab. Therefore, ab = 5. Therefore, area is 5/2. Hmm. So, this seems correct.But let me check with specific numbers. Let's take a and b such that a + b = sqrt(26), and a^2 + b^2 = 16. Let's compute a and b. Let me denote a = sqrt(26)/2 + t, b = sqrt(26)/2 - t. Then, a + b = sqrt(26), which is satisfied. Then, a^2 + b^2 = (sqrt(26)/2 + t)^2 + (sqrt(26)/2 - t)^2 = 2*(sqrt(26)/2)^2 + 2*t^2 = 2*(26/4) + 2t^2 = 13 + 2t^2. But according to Pythagoras, this should equal 16. So, 13 + 2t^2 = 16 => 2t^2 = 3 => t^2 = 3/2 => t = sqrt(3/2). Therefore, a = sqrt(26)/2 + sqrt(3/2) and b = sqrt(26)/2 - sqrt(3/2). Let's compute these values numerically.sqrt(26) is approximately 5.1, so sqrt(26)/2 ≈ 2.55. sqrt(3/2) ≈ 1.2247. So, a ≈ 2.55 + 1.2247 ≈ 3.7747, and b ≈ 2.55 - 1.2247 ≈ 1.3253. Then, checking a^2 + b^2: (3.7747)^2 ≈ 14.25, (1.3253)^2 ≈ 1.756, sum ≈ 16.006, which is close to 16, considering rounding errors. So, that checks out. Then, a*b ≈ 3.7747 * 1.3253 ≈ 5.0, which is 5. So, the area is indeed 5/2. Therefore, the answer is 5/2.Therefore, after all these checks, I think the area is 5/2.Final AnswerThe area of the triangle is boxed{dfrac{5}{2}}.</think>Given a right triangle with a perimeter of (4 + sqrt{26}) and a median length of 2 on the hypotenuse, we need to find the area of this triangle.1. Identify the hypotenuse: In a right triangle, the median to the hypotenuse is half the hypotenuse. Given the median is 2, the hypotenuse (c) is (2 times 2 = 4).2. Sum of the other two sides: The perimeter of the triangle is (4 + sqrt{26}). Therefore, the sum of the other two sides (a) and (b) is ((4 + sqrt{26}) - 4 = sqrt{26}). Thus, (a + b = sqrt{26}).3. Apply the Pythagorean theorem: For a right triangle, (a^2 + b^2 = c^2). Here, (c = 4), so (a^2 + b^2 = 16).4. Find the product (ab): Using the identity ((a + b)^2 = a^2 + 2ab + b^2): [ (sqrt{26})^2 = 16 + 2ab ] [ 26 = 16 + 2ab ] [ 2ab = 26 - 16 = 10 ] [ ab = 5 ]5. Calculate the area: The area of the triangle is (frac{1}{2}ab). Therefore, the area is (frac{1}{2} times 5 = frac{5}{2}).The area of the triangle is (boxed{dfrac{5}{2}}).]

🔑:Given a right triangle with a perimeter of 4+sqrt{26} and a median length of 2 on the hypotenuse, we are asked to find the area of this triangle. Let's denote the two legs of the right triangle as a and b, and the hypotenuse as c.First, we know that the median on the hypotenuse of a right triangle is equal to half of the hypotenuse. This gives us:[c = 2 times 2 = 4]Given that the perimeter of the right triangle is 4+sqrt{26}, we can write the equation for the perimeter as:[a + b + c = 4 + sqrt{26}]Substituting the value of c into this equation, we get:[a + b + 4 = 4 + sqrt{26}]Simplifying this, we find that:[a + b = sqrt{26}]From the Pythagorean theorem, we know that for a right triangle:[a^2 + b^2 = c^2]Substituting the value of c into this equation, we get:[a^2 + b^2 = 4^2 = 16]Now, we have a system of equations:[left{begin{array}{l}a + b = sqrt{26} a^2 + b^2 = 16end{array}right.]To find the area of the triangle, we need to calculate the product ab. We can use the identity (a + b)^2 = a^2 + 2ab + b^2 to express ab as:[ab = frac{1}{2}[(a + b)^2 - (a^2 + b^2)]]Substituting the known values into this equation, we get:[ab = frac{1}{2}[(sqrt{26})^2 - 16] = frac{1}{2}[26 - 16] = frac{1}{2}[10] = 5]Therefore, the area of the triangle, which is given by frac{1}{2}ab, is:[s_{triangle} = frac{1}{2} times 5 = frac{5}{2}]Hence, the area of the triangle is boxed{frac{5}{2}}.