Let P be a point on the ellipse x 2 9 + y 2 4 = 1 . Let the line passing through P and parallel to y - axis meet the circle x 2 + y 2 = 9 at point Q such that P and Q are on the same side of the x - axis. Then, the eccentricity of the locus of the point R on P Q such that P R : R Q = 4 : 3 as P moves on the ellipse, is:

Step-by-step solution

Given equation of ellipse and circle are x 2 9 + y 2 4 = 1 and x 2 + y 2 = 9 . ⇒ P ≡ 3 cos θ , 2 sin θ and Q ≡ 3 cos θ , 3 sin θ Let, R ≡ h , k ⇒ h = 3 cos θ , k = 18 7 sin θ ⇒ h 3 = cos θ , 7 k 18 = sin θ So, locus of R is given by, x 2 9 + 49 y 2 324 = 1 . ⇒ e = 1 − 324 49 × 9 ⇒ e = 117 21 ⇒ e = 13 7