Let P 2 3 7 , 6 7 , Q , R and S be four points on the ellipse 9 x 2 + 4 y 2 = 36 . Let P Q and R S be mutually perpendicular and pass through the origin. If 1 P Q 2 + 1 R S 2 = p q , where p and q are coprime, then p + q is equal to

Step-by-step solution

Given, P 2 3 7 ,   6 7 ,   Q ,   R and S be four points on the ellipse x 2 4 + y 2 9 = 1 Now, O P = r 1 = 2 3 7 2 + 6 7 2 = 48 7 where O is origin, Let P be r 1 cos θ ,   r 1 sin θ P lies on ellipse, so we get, r 1 2 cos 2 θ 4 + r 1 2 sin 2 θ 9 = 1 ⇒   cos 2 θ 4 + sin 2 θ 9 = 7 48       . . .   i Let R be - r 2 sin θ ,   r 2 cos θ as P Q   &   R S are perpendicular and pass through origin, So, r 2 2 sin 2