From any point P on the ellipse PN is drawn perpendicular to the major axis and produced at Q so that NQ equals to PS, where S is a focus. Then the locus of Q is

$\begin{array}{l}\mathrm{Let} \mathrm{P}=(\mathrm{a} \cos \mathrm{\theta },\mathrm{b} \sin \mathrm{\theta }) be any point on ellipse \\ \Rightarrow \mathrm{Let} \mathrm{Q}\left({\mathrm{x}}_1{\mathrm{y}}_1\right)\\ \therefore {\mathrm{x}}_1=\mathrm{acos} \mathrm{\theta },{\mathrm{y}}_1=SP\end{array}$
Now  $SP=|\mathrm{a}-\mathrm{aecos} \mathrm{\theta }|$
$\Rightarrow {\mathrm{y}}_1=\left|\mathrm{a}-{\mathrm{x}}_1\mathrm{e}\right|$

Locus of Q is 
$\begin{array}{l}\mathrm{ex}+\mathrm{y}-\mathrm{a}=0,\mathrm{ex}-\mathrm{y}-\mathrm{a}=0\\ \mathrm{Now} {\mathrm{S}}^1\mathrm{P}=|\mathrm{a}+\mathrm{aecos}\mathrm{\theta }| and {\mathrm{y}}_1=\left|\mathrm{a}+{\mathrm{ex}}_1\right|\\ \end{array}$
Locus of Q is $\mathrm{ex}+\mathrm{y}+\mathrm{a}=0,-\mathrm{ex}+\mathrm{y}-\mathrm{a}=0$