First slide

Ellipse

Question

Let the distance between a focus and the corresponding directrix of an ellipse be 8 and the eccentricity be 1/2.If the length of the minor axis is k, then $\sqrt{3} k / 2$ is __________.

Moderate

Solution

$\begin{array}{l} \frac{a}{e} - ae = 8 \\ or 2 a - \frac{a}{2} = 8 \\ or a = \frac{16}{3} \\ b^{2} = a^{2} (1 - e^{2}) = \frac{256}{9} (1 - \frac{1}{4}) = \frac{64}{3} \end{array}$
$∴$ Length of minor axis $= 2 b = \frac{16}{\sqrt{3}}$
$∴$ k = 8

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