Question

If $(5,12)$ and $(24,7)$ are the foci of an ellipse passing through the origin, then the eccentricity of the conic is

• A. $\frac{\sqrt{386}}{12}$
• B. $\frac{\sqrt{386}}{13}$
• C. $\frac{\sqrt{386}}{25}$
• D. $\frac{\sqrt{386}}{38}$

Answer 1

Answer: (D)

Let $S(5,12),S^{\prime }(24,7)$ be the two foci.
$P(0,0)$ is a point on the conic.
$SP=\sqrt{25+144}=13$
$S^{\prime }P=\sqrt{576+46}=\sqrt{625}=25$
$S{S}^{\prime }=\sqrt{(24-5{)}^2+(7-12{)}^2}$
$=\sqrt{19^2+5^2}=\sqrt{386}$
If the conic is an ellipse,
then $SP+S^{\prime }P=2a$ and $S{S}^{\prime }=2ae$
$\therefore e=\frac{S{S}^{\prime }}{SP+S^{\prime }P}$
$=\frac{\sqrt{386}}{13+25}=\frac{\sqrt{386}}{38}$