Question

$B$ is an extremity of the minor axis of an ellipse whose foci are $S$ and $S^{\prime }$. If $\angle SB{S}^{\prime }$ is a right angle, then the eccentricity of the ellipse is equals to:

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

Answer 1

Answer: (B)

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

Given : $\angle SB{S}^{\prime }$ is a right angle

$\therefore$ By Pythagoras theorem,

${S{S}^{\prime }}^2={SB}^2+{S^{\prime }B}^2$

$⟹(ae+ae{)}^2=(ae{)}^2+b^2+(ae{)}^2+b^2$

$⟹(2ae{)}^2=2(ae{)}^2+2b^2$

$⟹4(ae{)}^2=2(ae{)}^2+2b^2$

$⟹2(ae{)}^2=2b^2$

$⟹ae=b$

$⟹e=\frac{b}{a}$

But $e^2=1-\frac{b^2}{a^2}$

$⟹e^2=1-e^2$

$⟹2e^2=1$

$⟹e=\frac{1}{\sqrt{2}}$