Question

An ellipse has $OB$ as semi minor axis, $F$ and $F{}^{\prime }$ its focii and the angle $FBF{}^{\prime }$ is a right angle. Then the eccentricity of the ellipse is

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

Answer 1

Answer: (B)

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

$F^{\prime }(-ae,0)$, $F(ae,0)$ Slope of $BF=\frac{b}{ae}=m_1\left(say\right)$
Slope of $B{F}^{\prime }=\frac{b}{-ae}=m_2\left(say\right)$ now $m_1\times m_2=-1\Rightarrow \frac{b}{ae}\times \frac{b}{-ae}=-1$ $\Rightarrow b^2=a^2{e}^2$ $a^2-a^2{e}^2=a^2{e}^2$ $(\therefore e^2=1-\frac{b^2}{a^2})$
$\Rightarrow 1-e^2=e^2,2e^2=1,e=\pm \frac{1}{\sqrt{2}}$