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  4. In an ellipse, its foci and ends of its major axis are equally spaced. If the length of its semi-minor axis is 2√2, then the length of its semi-major axis is

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Q. In an ellipse, its foci and ends of its major axis are equally spaced. If the length of its semi-minor axis is $2\sqrt{2}$, then the length of its semi-major axis is

KVPYKVPY 2014

Solution:

Given, in ellipse
$A'S' = SS' = SA$
image
$\therefore SS' = \frac{1}{3} AA' $
$\Rightarrow 2ae = \frac{1}{3}(2a)$
$\Rightarrow e = \frac{1}{3}$
Also given, $b = 2 \sqrt{2}$
$e^2 = 1 - \frac{b^2}{a^2}$
$\Rightarrow \frac{1}{9} = 1 - \frac{8}{a^2}$
$\Rightarrow \frac{8}{a^2} = 1 - \frac{1}{9} = \frac{8}{9}$
$\Rightarrow a^2 = 9$
$ \therefore a = 3$