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The slope of the tangent at (x, y) to a curve passing through (2, 1) is (x2 + y2)/2xy, then the equation of the curve is
- a)2 (x2 - y2) = 3 x
- b)2 (x2 - y2) = 6 y
- c)x (x2 - y2) = 6
- d)x (x2 + y2) = 10
Correct answer is option 'A'. Can you explain this answer?
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The slope of the tangent at (x, y) to a curve passing through (2, 1) i...
Solution:
To find the equation of the curve, we need to integrate the given slope expression with respect to x. Let's solve step by step.
Step 1: Find the derivative of y with respect to x using the given slope expression.
Given slope expression: slope = (x^2 - y^2) / 2xy
Using the derivative formula, we have:
dy/dx = (x^2 - y^2) / 2xy
Step 2: Rewrite the derivative equation in terms of differentials.
Multiplying both sides by 2xy, we get:
2xy dy = (x^2 - y^2) dx
Step 3: Integrate both sides of the equation.
Integrating both sides, we have:
∫2xy dy = ∫(x^2 - y^2) dx
Step 4: Evaluate the integrals.
Integrating the left side:
∫2xy dy = x^2y + C1, where C1 is the constant of integration.
Integrating the right side:
∫(x^2 - y^2) dx = x^3/3 - xy^2/3 + C2, where C2 is the constant of integration.
Step 5: Equate the two results and solve for y.
Equating the results, we have:
x^2y + C1 = x^3/3 - xy^2/3 + C2
Simplifying the equation:
3x^2y + 3C1 = x^3 - xy^2 + 3C2
Rearranging the terms:
x^3 - 3x^2y - xy^2 = 3C2 - 3C1
Step 6: Use the given point (2, 1) to find the value of the constants.
We are given that the curve passes through the point (2, 1). Substituting x = 2 and y = 1 into the equation, we have:
2^3 - 3(2^2)(1) - 2(1^2) = 3C2 - 3C1
Simplifying the equation:
8 - 12 - 2 = 3C2 - 3C1
-6 = 3C2 - 3C1
Step 7: Substitute the values of the constants back into the equation.
Substituting the value of -6 for 3C2 - 3C1, we have:
x^3 - 3x^2y - xy^2 = -6
Simplifying the equation further, we get:
x^3 - 3x^2y - xy^2 + 6 = 0
Step 8: Final equation of the curve.
The equation of the curve is:
x^3 - 3x^2y - xy^2 + 6 = 0
Comparing this equation with the given options, we can see that option 'A' matches the equation obtained.
Therefore, the correct answer is option 'A': 2(x^2 - y^2) = 3x.
To find the equation of the curve, we need to integrate the given slope expression with respect to x. Let's solve step by step.
Step 1: Find the derivative of y with respect to x using the given slope expression.
Given slope expression: slope = (x^2 - y^2) / 2xy
Using the derivative formula, we have:
dy/dx = (x^2 - y^2) / 2xy
Step 2: Rewrite the derivative equation in terms of differentials.
Multiplying both sides by 2xy, we get:
2xy dy = (x^2 - y^2) dx
Step 3: Integrate both sides of the equation.
Integrating both sides, we have:
∫2xy dy = ∫(x^2 - y^2) dx
Step 4: Evaluate the integrals.
Integrating the left side:
∫2xy dy = x^2y + C1, where C1 is the constant of integration.
Integrating the right side:
∫(x^2 - y^2) dx = x^3/3 - xy^2/3 + C2, where C2 is the constant of integration.
Step 5: Equate the two results and solve for y.
Equating the results, we have:
x^2y + C1 = x^3/3 - xy^2/3 + C2
Simplifying the equation:
3x^2y + 3C1 = x^3 - xy^2 + 3C2
Rearranging the terms:
x^3 - 3x^2y - xy^2 = 3C2 - 3C1
Step 6: Use the given point (2, 1) to find the value of the constants.
We are given that the curve passes through the point (2, 1). Substituting x = 2 and y = 1 into the equation, we have:
2^3 - 3(2^2)(1) - 2(1^2) = 3C2 - 3C1
Simplifying the equation:
8 - 12 - 2 = 3C2 - 3C1
-6 = 3C2 - 3C1
Step 7: Substitute the values of the constants back into the equation.
Substituting the value of -6 for 3C2 - 3C1, we have:
x^3 - 3x^2y - xy^2 = -6
Simplifying the equation further, we get:
x^3 - 3x^2y - xy^2 + 6 = 0
Step 8: Final equation of the curve.
The equation of the curve is:
x^3 - 3x^2y - xy^2 + 6 = 0
Comparing this equation with the given options, we can see that option 'A' matches the equation obtained.
Therefore, the correct answer is option 'A': 2(x^2 - y^2) = 3x.
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The slope of the tangent at (x, y) to a curve passing through (2, 1) is (x2 + y2)/2xy, then the equation of the curve isa)2 (x2 - y2) = 3 xb)2 (x2 - y2) = 6 yc)x (x2 - y2) = 6d)x (x2 + y2) = 10Correct answer is option 'A'. Can you explain this answer?
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The slope of the tangent at (x, y) to a curve passing through (2, 1) is (x2 + y2)/2xy, then the equation of the curve isa)2 (x2 - y2) = 3 xb)2 (x2 - y2) = 6 yc)x (x2 - y2) = 6d)x (x2 + y2) = 10Correct answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The slope of the tangent at (x, y) to a curve passing through (2, 1) is (x2 + y2)/2xy, then the equation of the curve isa)2 (x2 - y2) = 3 xb)2 (x2 - y2) = 6 yc)x (x2 - y2) = 6d)x (x2 + y2) = 10Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The slope of the tangent at (x, y) to a curve passing through (2, 1) is (x2 + y2)/2xy, then the equation of the curve isa)2 (x2 - y2) = 3 xb)2 (x2 - y2) = 6 yc)x (x2 - y2) = 6d)x (x2 + y2) = 10Correct answer is option 'A'. Can you explain this answer?.
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