First slide

Ellipse

Question

$A set of lines x + y - 2 + λ (2 x + y - 3) = 0 represent incident rays on an ellipse s = 0$ $and 2 x + 3 y - 23 + μ (2 x + y - 3) = 0 represent the set of reflective rays from the$ $same ellipse S = 0, (λ, μ \in ℝ) . If P (3, 7) is a point on S = 0 and if the eccentricity of$ $the ellipse is \sqrt{A} - \sqrt{B} (A, B \in N) G.C.D (A, B) = 1 then A - B equals$

Difficult

Solution

$\begin{matrix} The foci of the ellipse are points of intersections of family of lines given \\ S (1, 1), S^{'} (- \frac{7}{2}, 10) \\ The eccentricity of the ellipse is e \\ e = \frac{S S^{'}}{S P + S^{'} P} \\ = \frac{\frac{1}{2} \sqrt{405}}{2 \sqrt{10} + \frac{1}{2} \sqrt{405}} \\ = \sqrt{41} - \sqrt{32} \\ 41 - 32 = 9 \end{matrix}$

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