If C P and C D are conjugate semi-diameters of the ellipse x 2 a 2 + y 2 b 2 = 1 , then, the locus of the mid-point of P D is

Step-by-step solution

Let ( h , k ) be the midpoint of P D . Then, h = a cos θ - a sin θ 2 ⇒ 2 h = a cos θ - a sin θ ⇒ 2 h a = cos θ - sin θ     . . . 1 And, k = b sin θ + b cos θ 2 ⇒ 2 k = b sin θ + b cos θ ⇒ 2 k b = sin θ + cos θ     . . . 2 Squaring and adding 1 and 2 , we get 4 h 2 a 2 + 4 k k b 2 = cos θ - sin θ 2 + sin θ + cos θ 2 ⇒ 4 h 2 a 2 + 4 k 2 b 2 = 2 ⇒ h 2 a 2 + k 2 b 2 = 1 2 Hence, the lo