If `sqrt(1-x^6)+sqrt(1-x^6)=a(x^3-y^3),` then prove that `(dy)/(dx)=(x^2)/(y^2)sqrt((1-y^6)/(1-x^6))`

Options

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)`

Answer

D

Explanation

Put `x^(3)= sin theta, y^(3) = sin phi, ` then
`(cos theta+cos phi)=a (sin theta-sin phi)`
`rArr 2 cos((theta+phi)/(2))cos((theta-phi)/(2))=2a cos ((theta+phi)/(2))sin((theta-phi)/(2))`
`rArr cot((theta-phi)/(2))=a`
`rArr (theta-phi)/(2)=cot^(-1)a rArr sin^(-1)x^(3)-sin^(-1)y^(3)=2cot^(-1)a`
`therefore (3x^(2))/(sqrt((1-x^(6))))-(3y^(2))/(sqrt((1-y^(6))))(dy)/(dx)=0rArr (dy)/(dx)=(x^(2))/(y^(2))sqrt(((1-y^(6))/(1-x^(6))))`
`therefore f(x, y)=(x^(2))/(y^(2))`