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 ,…

Question

Let E 1 and E 2 be two ellipse whose centers are at the origin. The major axes of E 1 a n d &#160; 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 , &#160; E 1 and E 2 at P , &#160; Q and R , respectively. Suppose that P Q = P R = &#160; 2 2 3 . If &#160; e 1 and e 2 are the eccentricities of E 1 and E 2 , respectively, then the correct expression(s) is(are)

Aicharya · Accepted Answer

Let Q be ( x 1 , &#160; 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 &#160; &#8801; ( 1,2 ) So R and Q will be 1 &#160; &#177; 2 2 3 cos &#8289; 3 &#960; 4 , &#160; 2 &#177; 2 2 3 sin &#8289; 3 &#960; 4 &#8658; &#160; &#160; Q &#160; 5 3 &#160; , 4 3 &#160; a n d &#160; 1 3 &#160; , 8 3 Let Q lies on x 2 &#160; a 2 + &#160; 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 , &#160; b 2 = 4 &#160; &#8658; e 1 = 1 5 And R lies on x 2 a 1 2 + y 2 b 1 2 = 1 So tangent at x 3 a 1 2 + 8 y 3 b 1 2 = 3 Comparing with x + y = 3 &#8658; &#160; &#160; &#160; a 1 2 = 1 , &#160; b 1 2 = 8 &#160; &#160; &#160; &#8658; &#160; &#160; e 2 = 7 8 &#8658; &#160; &#160; &#160; e 1 2 + e 2 2 = 43 40 &#160; a n d &#160; e 1 e 2 = 7 2 10

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…

Solution

Let E 1 and E 2 be two ellipse whose centers are at the origin. The major axes of E 1 a n d &#160; E 2 lie…

Solution

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…