Question

If $(5,12)$ and $(12,16)$ are the foci of an ellipse passing through $(0,0),$ then the eccentricity of the ellipse is $\frac{\sqrt{65}}{33}$

Answer 1

Let $P$ be $(0,0)$ and

$S$ and $S^{\prime }$ be $(5,12)$ and $(12,16)$ respectively

$S{S}^{\prime }=2ae$

$2ae=\sqrt{(5-12{)}^2+(16-12{)}^2}$

$\sqrt{7^2+4^2}=\sqrt{49+16}=\sqrt{65}$

$4a^2{e}^2=65$

$\sqrt{5^2+{12}^2}+\sqrt{{12}^2+{16}^2}=2a$

$2a=\sqrt{{13}^2}+\sqrt{{20}^2}=13+20=33$

$a=\frac{33}{2}$

$4a^2{e}^2=65$

$e^2=\frac{65}{4}\times \frac{1}{a^2}=\frac{65}{4}\times \frac{4}{{33}^2}$

$e=\frac{\sqrt{65}}{33}$