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Question

If 1-x6+1-y6=a(x3-y3) and dydx=f(x,y)1-y61-x6, then


A

f(x,y)=yx

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B

f(x,y)=2yx

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C

f(x,y)=y2x2

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D

f(x,y)=x2y2

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Solution

The correct option is D

f(x,y)=x2y2


Explanation for the correct option:

Step 1. Put x3=cosθ and y3=cosɸ in given equation, we get

1-x6+1-y6=a(x3-y3)

1cos2θ+1cos2ɸ=a(cosθcosɸ)

sinθ+sinɸ=a(cosθcosɸ)

2sin[θ+ɸ]2cos[θɸ]2=2asin[θ+ɸ]2sin[θɸ]2

cot[θɸ]2=a

[θɸ]=2cot-1(-a) (Constant)

Step 2. Differentiate the given equation,

1-x6+1-y6=a(x3-y3)

3x21x6+3y21y6dydx=0

dydx=1y61x6x2y2

Comparing this with dydx=f(x,y)1-y61-x6

f(x,y)=x2y2

Hence , Option ‘D’ is Correct.


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