if ` x = a cos^3 theta sin^2 theta and y = a cos^2 theta sin^3 theta` and `(x^2 + y^2)^p/(xy)^q` is independent of `theta`, then (A) `4p=5q` (B) `5p=4q` (C) `p+q=9` (D) `pq=20`

Explanation

We have,
`x^(2)+y^(2)=a^(2)cos^(6)thetasin^(4)theta+a^(2)sin^(6)thetacos^(4)theta`
`=a^(2)sin^(4)thetacos^(4)theta(cos^(2)theta+sin^(2)theta)`
`=a^(2)sin^(4)thetacos^(4)theta` (`:cos^(2)theta+sin^(2)theta=1)`
`xy=a^(2)cos^(5)thetasin^(5)theta`
`:. ((x^(2)+y^(2))^(p))/((xy)^(q))=(a^(2)(sin^(4)thetacos^(4)theta)^(p))/(a^(2q)(cos^(5)thetasin^(5)theta)^(q))`
`=(a^(2p)(sinthetacostheta)^(4p))/(a^(2q)(sinthetacostheta)^(5))`
But since `((x^(2)+y^(2))^(p))/((xy)^(q))` is independent od `theta`
`:.4p=5p`
`(p)/(q)=(5)/(4)rArrp :q =5:4`