If ` x^(2//3) + y^(2//3) = a^(2//3) , "then" (dy)/(dx)` is equal to

Options

A. `-root(3)((y)/(x))`
B. `-root(3)(y//x)`
C. `(y)/(x) `
D. None of these

Answer

A

Explanation

Let ` x = a cos^(3) theta , y = a sin^(3) theta `. Then
` x^((2)/(3))+ y^((2)/(3))= (a cos^(3) theta)^((2)/(3)) + (a sin^(3) theta )^((2)/(3))`
`a^((2)/(3))[(cos^(2) theta + (sin^(2) theta )] = a^((2)/(3))`
Thus , ` x = a cos^(3) theta , y = a sin^(3) theta ` is parametric equation of
` x^((2)/(3)). + y^((2)/(3)) = a^((2/(3))`
Now , ` (dx)/(dtheta) = - 3 a cos^(2) theta sin theta and (dy)/(dtheta) = 3 a sin^(2) theta cos theta `
`(dx)/(d theta) = - 3 a cos^(2) theta sin theta and (dy)/(d theta ) = 3 a sin^(2) theta cos theta `ltbr. `rArr (dy)/(dx) = ((dy)/(d theta))/((dx)/(d theta)) = (3 a sin^(2) theta cos theta )/(-3a cos^(2) theta cos^(2) sin theta)`
` = tan theta = - root(3)((y)/(x))`