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)`
`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))`