Question

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

• A. $4$
• B. $2\sqrt{3}$
• C. $\sqrt{10}$
• D. $3$

Answer 1

Answer: (D)

Given, in ellipse
$A^{\prime }{S}^{\prime }=S{S}^{\prime }=SA$

$\therefore S{S}^{\prime }=\frac{1}{3}A{A}^{\prime }$
$\Rightarrow 2ae=\frac{1}{3}(2a)$
$\Rightarrow e=\frac{1}{3}$
Also given, $b=2\sqrt{2}$
$e^2=1-\frac{b^2}{a^2}$
$\Rightarrow \frac{1}{9}=1-\frac{8}{a^2}$
$\Rightarrow \frac{8}{a^2}=1-\frac{1}{9}=\frac{8}{9}$
$\Rightarrow a^2=9$
$\therefore a=3$