Question

S and S' foci of an ellipse. B is one end of the minor axis. If $\angle SB{S}^{\prime }$ is a right angled isosceles triangle, then e$=?$

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

Answer 1

The correct option is A $\frac{1}{\sqrt{2}}$

We have

$S=(ae,0)S^{\prime }(-ae,0)andB=(0,b)$

Since it is given that $\angle SB{S}^{\prime }={90}^o$

Slope of SB$\times$ Slope of S'B$=-1$

$\left(\frac{b-0}{0-ae}\right)\times \left(\frac{b-0}{b+ae}\right)=-1$

$\left(\frac{-b}{ae}\right)\left(\frac{b}{ae}\right)=-1$

$b^2=a^2{e}^2$

But, $b^2=a^2(1-e^2)$

So,

$a^2(1-e^2)=a^2{e}^2$

$1-e^2=e^2$

$2e^2=1$

$e^2=\frac{1}{2}$

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