Question

The ellipse cuts $x$ axis at $A$ and $y$ axis at $B$ and the line joining the focus $S$ and $B$ makes an angle $\frac{3\pi }{4}$ with $x$-axis, then the eccentricity of the ellipse is:

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

Answer 1

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

In Triangle BOS,

$OB=b$

$OS=ae$ (e=eccentricity of the Ellipse)

$\angle BSO=\frac{\pi }{4}$

$\therefore \frac{OB}{OS}=\tan \frac{\pi }{4}$

or, $OB=OS$

or, $b=ae$

Squaring both sides we get,

$b^2={\left(ae\right)}^2$

And we know,

$b^2=a^2-{\left(ae\right)}^2$

$\therefore 2{\left(ae\right)}^2=a^2$

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

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