Let E1 and E2 be two ellipses whose centers are at the origin. The major axes of E1 and E2 lie along

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Let E1 and E2 be two ellipses whose centers are at the origin. The major axes of E1 and E2 lie along the x-axis and the y-axis, respectively. Let S be the circle x2 + (y &#8722; 1)2 = 2. The straight line x + y = 3 touches the curves S, E1 and E2 at P,Q and R, respectively. Suppose that PQ = PR = If e1 and e2 are the eccentricities of E1 and E2, respectively, then the correct expression(s) is(are)

Accepted Answer

Let E1 : (a > b)& E2 : (c < d)& S : x2 + (y - 1)2 = 2& tangent to E1, E2 & S is x + y = 3Now, point of contact of S & tangent is (x1, y1)Let x = X & y - 1 = Y&#8756; X2 + Y2 = 2& X + Y = 2Let (X1, Y1) be point of contact.&#8756; XX1 + YY1 = 2&#8756; X1 = 1 & Y1 = 1&#8756; x1 = 1 & y1 = 2Now, parametric equation of x + y = 3is &#8658; &#8756; P &#8801; (1, 2), Q &#8801; & R &#8801; Now, equation tangent at Q on ellipse E1 Comparing it with x + y = 3&#8756; a2 = 5 & b2 = 4 Now, e12 = 1 - Similarly, e22 = &#8756; e12e22 = &#8658; e1e2 = e12 + e22 = ; |e12 - e22 | = &#8756; (A) & (B)

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