First slide

Ellipse

Question

$P and Q are the foci of the ellipse \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 and B is an end of the minor axis. If P B Q is an equilateral triangle,$
$then the eccentricity of the ellipse is$

Moderate

A

$\frac{1}{\sqrt{2}}$
B

$\frac{1}{3}$
C

$\frac{1}{2}$
D

$\frac{\sqrt{3}}{2}$

Solution

$\begin{array}{l} We have P Q = B P \\ or 2 a e = \sqrt{a^{2} e^{2} + b^{2}} = \sqrt{a^{2}} = a \\ or e = \frac{1}{2} \end{array}$

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