First slide

Ellipse

Question

$A pair of tangents S A and S B are drawn to parabola y^{2} = 8 x from a foci S of ellipse \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (with a > b),$
$such that Δ S A B is an equilateral triangle. If the tangents S A and S B pass through the extremities of minor axis of$
$ellipse, then 4 e^{2} equals (where e is eccentricity of ellipse)$

Moderate

Solution

$\begin{array}{l} \frac{b}{a e} = \frac{1}{\sqrt{3}} \\ \Rightarrow \frac{b^{2}}{a^{2}} = \frac{e^{2}}{3} \\ \Rightarrow 1 - e^{2} = \frac{e^{2}}{3} \\ \Rightarrow 4 e^{2} = 3 \end{array}$

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