$\text{If (5 , 12) and (24, 7) arc the foci of an ellipse passing through the origin, then the eccentricity of the ellipse is}$

$\text{ The two foci are }S(5,12)\text{ and }{S}^{\mathrm{\prime }}(24,7)\text{ and the ellipse }$

$\text{ passes through origin }O,\text{ then }$

$\begin{array}{l}SO=\sqrt{25+144}=13;S^{\mathrm{\prime }}O=\sqrt{576+49}=25\\ \text{ and }S{S}^{\mathrm{\prime }}=\sqrt{386}\end{array}$

$\begin{array}{l}\text{ If a conic be an ellipse, then }SO+S^{\mathrm{\prime }}O=2a\text{ and }S{S}^{\mathrm{\prime }}=2ae\\ \therefore e=\frac{S{S}^{\mathrm{\prime }}}{S^{\mathrm{\prime }}O+SO}=\frac{\sqrt{386}}{38}\end{array}$