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If sqrt((1-x^(6)))+sqrt((1-y^(6)))=a(x^(...

If `sqrt((1-x^(6)))+sqrt((1-y^(6)))=a(x^(3)-y^(3))` and `(dy)/(dx)=f(x,y) sqrt(((1-y^(6))/(1-x^(6))))`, then

A

`f(x,y) = y//x`

B

`f(x,y) =y^2//x^2`

C

`f(x,y) = 2y^2//x^2`

D

`f(x,y) = x^2 //y^2`

Text Solution

Verified by Experts

The correct Answer is:
D
Put ` x^(3) = sin theta , y^(3) = sin phi` , then
` (cos theta + cos phi ) = a( sin theta - sin phi)`
`implies 2 cos((theta + phi )/(2)) cos ((theta-phi)/(2)) = 2a cos((theta + phi)/(2)) sin((theta - phi)/(2))`
` implies cot ((theta - phi)/(2)) =a`
`implies (theta - phi)/(2) = cot^(-1) a implies sin^(-1) x^(3) - sin^(-1)y^(3) = 2 cot^(-1)a`
`:.(3x^(2))/(sqrt(1-x^(6)))-(3y^(2))/(sqrt(1-y^(6)))(dy)/(dx) = 0 implies (dy)/(dx) = (x^(2))/(y^(2)) sqrt(((1-y^(6))/(1-x^6)))`
`:. f(x,y) = (x^(2))/(y^(2))`
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