Question

An ellipse has $OB$ as semi-minor axis $F$and $F\text{'}$ its foci and the angle$FBF\text{'}$ is a right angle. Then, the eccentricity of the ellipse is

• A. $\frac{1}{\sqrt{3}}$
• B. $\frac{1}{4}$
• C. $\frac{1}{2}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

The correct option is D

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

Step 1. Given data

An ellipse has $OB$ as semi-minor axis $F$and $F\text{'}$ its foci and the angle$FBF\text{'}$ is a right angle.

Step 2. Finding the eccentricity, $e$

Since, $F$ and $F\text{'}$ are foci of an ellipse whose coordinates are$(ae,0)$ and $(-ae,0)$ respectively and coordinates of $B$ are $(0,b)$.

Slope of $BF=\frac{b}{ae}$

Slope of $BF\text{'}=-\frac{b}{ae}$

$\begin{array}{rcl}\angle FBF& =& 90°\\ \left(\frac{-b}{ae}\right)\left(\frac{b}{ae}\right)& =& -1\\ b^2& =& a^2{e}^2\\ e^2& =& 1-\left(\frac{b^2}{a^2}\right)\\ & =& 1–\left(\frac{a^2{e}^2}{a^2}\right)\\ 2e^2& =& 1\\ e& =& \frac{1}{\sqrt{2}}\end{array}$

Hence, the correct option is $\left(D\right)$.