Question

If p is a parameter of positive numbers, show that all members of the family of ellipses x²/ (a² + p) + y² / (b² + p) = 1 have the same foci.

Answer 1

If the equation of an ellipse is $\frac{x^2}{m^2}+\frac{y^2}{n^2}=1,$ the the foci are $(-c,0)$ and $(c,0),$ where $c=\sqrt{m^2-n^2}.$ In our case,

$c=\sqrt{m^2-n^2}=\sqrt{(\sqrt{a^2+p})^2-(\sqrt{b^2+p})^2}\\ =\sqrt{a^2+p-(b^2+p)}=\sqrt{a^2-b^2}.$

We conclude that the foci are $(-\sqrt{a^2-b^2},0)$ and $(\sqrt{a^2-b^2},0).$ Since the foci do not depend on the parameter $p,$ the family of ellipses $\frac{x^2}{a^2 + p} + \frac{y^2}{b^2 + p} = 1$ have the same foci.