Question

The foci of an ellipse are $S(-2,-3),S^{\prime }(0,1)$ and its $e=\frac{1}{\sqrt{2}}$ then the directrix corresponding to the focus $S^{\prime }$ is:

• A. $x+2y-5=0$
• B. $x+2y-9=0$
• C. $x+2y-11=0$
• D. none of these

Answer 1

The correct option is D none of these

Given, foci of an ellipse are $S(-2,-3)$ and $S^{\prime }(0,1)$ and eccentricity $e=\frac{1}{\sqrt{2}}$
We know that distance between foci is $2ae=\sqrt{{(0-2)}^2+{(-3-1)}^2}=\sqrt{20}⟹a=\sqrt{10}$
Directrix and line joining foci are perpendicular to each other,

$\therefore$ slope of directrix is $(-\frac{0+2}{1+3})=-\frac{1}{2}$ and it will be parallel to line at point $S^{\prime }$ with this slope.
Equation of line at $S^{\prime }$ is $\frac{y-1}{x-0}=-\frac{1}{2}⟹x+2y-2=0$, then equation of directrix will be $x+2y+k=0$ and its distance from point $S^{\prime }$ is $a(\frac{1}{e}-e)=\sqrt{5}=\frac{0+2\left(1\right)+k}{\sqrt{5}}⟹k=3$
$\therefore$ Equation of directrix of ellipse is $2y+x+3=0$