First slide

Ellipse

Question

$Let S and S^{'} be two foci of the ellipse \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 . If a circle described on S S^{'} as diameter intersects the ellipse$
$at real and distinct points, then the eccentricity e of the ellipse satisfies$

Moderate

A

$e = 1 / \sqrt{2}$
B

$e \in (1 / \sqrt{2}, 1)$
C

$e \in (0, 1 / \sqrt{2})$
D

$None of these$

Solution

$The radius of the circle having SS'as diameter is r: ae. If it cuts an ellipse, then$
$r > b$
$\begin{array}{ll} or a e > b \\ or e^{2} > \frac{b^{2}}{a^{2}} \end{array}$
$\begin{array}{ll} or e^{2} > 1 - e^{2} \\ or e^{2} > \frac{1}{2} \end{array}$
$\begin{array}{l} or e > \frac{1}{\sqrt{2}} \\ or e \in (\frac{1}{\sqrt{2}}, 1) \end{array}$

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