Let P be a point on the ellipse x 2 a 2 + y 2 b 2 = 1 ,   0 < b < a . Let the line parallel to y -axis passing through P meets the circle x 2 + y 2 = a 2 at the point Q such that P and Q are on the same side of x -axis. For two positive real numbers r and s , find the locus of the point R on P Q such that P R : R Q = r : s , as P varies over the ellipse.

Step-by-step solution

Given, P R R Q = r s ⇒ α - b sin θ a sin θ - α = r s ⇒ α s - b sin θ s = r a sin θ - α r ⇒ α s + α r = r a sin θ + b sin θ · s ⇒ α s + r = sin θ r a + b s ⇒ α = sin θ r a + b s r + s Let the coordinate of R be h , k . ⇒   h = a cos θ and k = α = a r + b s sin θ r + s ⇒ cos θ = h a ,   sin θ = k r + s a r + b s On squaring and adding, we get sin 2 θ +