The locus of extremities of the latus-rectum of the family of ellipse ${\mathrm{b}}^2{\mathrm{x}}^2+{\mathrm{a}}^2{\mathrm{y}}^2={\mathrm{a}}^2{\mathrm{b}}^2\text{ is }(\mathrm{b}\in \mathrm{R})$

 $$\begin{gathered} Given ellipse \frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b) \\ Locus of extremities of the latus recturn \left(h,k_{ }\right)=\left(\pm \mathrm{ae},\pm \frac{{\mathrm{b}}^2}{\mathrm{a}}\right) \\ \therefore k=\pm \frac{b^2}{a} \\ h=\pm ae \\ \Rightarrow h^2=a^2{e}^2 \\ \Rightarrow h^2=a^2-b^2 \\ \Rightarrow h^2=a^2\pm ak \\ \Rightarrow h^2\pm ak=a^2 \\ locus of (h,k) is x^2\pm ay=a^2 \end{gathered}$$