Let E 1 and E 2 be two ellipse whose centers are at the origin. The major axes of E 1 a n d E 2 lie along the x - axis and the y - axis, respectively. Let S be the circle x 2 + y - 1 2 = 2 . The straight line x + y = 3 touches the curves S , E 1 and E 2 at P , Q and R , respectively. Suppose that P Q = P R = 2 2 3 . If e 1 and e 2 are the eccentricities of E 1 and E 2 , respectively, then the correct expression(s) is

Step-by-step solution

Let Q be ( x 1 ,   y 1 ) So equation tangent at Q will be x x 1 + y y 1 - y + y 1 - 1 = 0 Comparing with x + y = 3 x 1 y 1   ≡ ( 1,2 ) So R and Q will be 1   ± 2 2 3 cos ⁡ 3 π 4 ,   2 ± 2 2 3 sin ⁡ 3 π 4 ⇒     Q   5 3   , 4 3   a n d   1 3   , 8 3 Let Q lies on x 2   a 2 +   y 2 b 2 = 1 So tangent at P is 5 x 3 a 2 + 4 y 3 b 2 = 1 Comparing with x + y = 3 a 2 = 5 ,   b 2 = 4   ⇒ e 1 = 1 5 And