If (5, 12) and (24, 7) are the foci of a hyperbola passing though the origin then the eccentricity of the hyperbola is

Let $S(5, 12)$ and $S^{\mathrm{\prime }}(24,7)$ be two foci and

$P(0, 0)$ be a point on the conic

then  $\begin{array}{l}SP=\sqrt{25+144}=\sqrt{169}=13\\ S^{\mathrm{\prime }}P=\sqrt{(24{)}^2+7^2}=\sqrt{625}=25\\ S{S}^{\mathrm{\prime }}=\sqrt{(24-5{)}^2+(7-12{)}^2}=\sqrt{{19}^2+5^2}\\ =\sqrt{386}\end{array}$

since the conic is a hyperbola, $S^{\mathrm{\prime }}P-SP=2a$, the length of transverse axis 

and$S{S}^{\mathrm{\prime }}=2ae,e$  being the eccentricity.

$\Rightarrow e=\frac{S{S}^{\mathrm{\prime }}}{S^{\mathrm{\prime }}P-SP}=\frac{\sqrt{386}}{12}$