If `sqrt(1-x^(2n))+sqrt(1-y^(2n))=a^n(x^n-y^n),` prove that `y^(n-1)*sqrt(1-x^(2n)) dy = x^(n-1)sqrt(1-y^(2n)) dx.`

Options

A. `(x^(n-1))/(y^(n-1))`
B. `(y^(n-1))/(x^(n-1))`
C. `(x)/(y)`
D. 1

Answer

A

Explanation

Put ` x^(x) = sin theta " and " y^(n) = 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 theta - phi = 2 cot ^(-1) a `
or ` sin^(-1) x^(n) - sin^(-1) y^(n) = 2 cot^(-1) a `
`(nx^(n-1))/(sqrt(1 -x^(2n))) - (ny^(n-1))/(sqrt((1 -y^(2n))) ) (dy)/(dx) = 0 `
` therefore sqrt(((1-x^(2n))/(1 -y^(2n)))) (dy)/(dx) = (x^(n-1))/(y^(n-1))`