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The value of α ∈ R for which the curves x2 + αy2 = 1 and y = x2 intersect orthogonally is
- a)-2
- b)-1/2
- c)1/2
- d)2
Correct answer is option 'D'. Can you explain this answer?
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The value of R for which the curves x2 + y2 = 1 and y = x2 intersect...
To find the value of R for which the curves x^2 - y^2 = 1 and y = x^2 intersect orthogonally, we need to determine the slope of the curves at their point of intersection.
1. Find the slopes of the curves:
- Differentiate the equations with respect to x to find the slopes of the curves.
- Differentiating x^2 - y^2 = 1 gives 2x - 2yy' = 0, where y' denotes dy/dx.
- Differentiating y = x^2 gives y' = 2x.
2. Substitute the values of x and y' from the first equation into the second equation:
2x - 2yy' = 0
2x - 2y(2x) = 0
2x - 4xy = 0
2x(1 - 2y) = 0
3. Solve for x:
2x = 0 or 1 - 2y = 0
x = 0 or y = 1/2
4. Substitute the values of x and y into the original equations to find the corresponding values of R:
For x = 0, y = ±1, and R = 1.
For y = 1/2, x^2 - (1/2)^2 = 1
x^2 - 1/4 = 1
x^2 = 5/4
x = ±√(5/4)
R = (x^2 - y^2)/(2x) = ((5/4) - (1/2)^2)/(2√(5/4)) = (5/4 - 1/4)/(2√(5/4)) = 1/2√(5/4) = 2/√5 = (2/√5) * (√5/√5) = 2√5/5.
Thus, the values of R for which the curves intersect orthogonally are 1 and 2√5/5. The correct answer is option D.
1. Find the slopes of the curves:
- Differentiate the equations with respect to x to find the slopes of the curves.
- Differentiating x^2 - y^2 = 1 gives 2x - 2yy' = 0, where y' denotes dy/dx.
- Differentiating y = x^2 gives y' = 2x.
2. Substitute the values of x and y' from the first equation into the second equation:
2x - 2yy' = 0
2x - 2y(2x) = 0
2x - 4xy = 0
2x(1 - 2y) = 0
3. Solve for x:
2x = 0 or 1 - 2y = 0
x = 0 or y = 1/2
4. Substitute the values of x and y into the original equations to find the corresponding values of R:
For x = 0, y = ±1, and R = 1.
For y = 1/2, x^2 - (1/2)^2 = 1
x^2 - 1/4 = 1
x^2 = 5/4
x = ±√(5/4)
R = (x^2 - y^2)/(2x) = ((5/4) - (1/2)^2)/(2√(5/4)) = (5/4 - 1/4)/(2√(5/4)) = 1/2√(5/4) = 2/√5 = (2/√5) * (√5/√5) = 2√5/5.
Thus, the values of R for which the curves intersect orthogonally are 1 and 2√5/5. The correct answer is option D.
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The value of R for which the curves x2 + y2 = 1 and y = x2 intersect orthogonally isa)-2b)-1/2c)1/2d)2Correct answer is option 'D'. Can you explain this answer?
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The value of R for which the curves x2 + y2 = 1 and y = x2 intersect orthogonally isa)-2b)-1/2c)1/2d)2Correct answer is option 'D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about The value of R for which the curves x2 + y2 = 1 and y = x2 intersect orthogonally isa)-2b)-1/2c)1/2d)2Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The value of R for which the curves x2 + y2 = 1 and y = x2 intersect orthogonally isa)-2b)-1/2c)1/2d)2Correct answer is option 'D'. Can you explain this answer?.
The value of R for which the curves x2 + y2 = 1 and y = x2 intersect orthogonally isa)-2b)-1/2c)1/2d)2Correct answer is option 'D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about The value of R for which the curves x2 + y2 = 1 and y = x2 intersect orthogonally isa)-2b)-1/2c)1/2d)2Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The value of R for which the curves x2 + y2 = 1 and y = x2 intersect orthogonally isa)-2b)-1/2c)1/2d)2Correct answer is option 'D'. Can you explain this answer?.
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