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.
$S P=\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 $S P+S^{\prime} P=2 a$ and $S S^{\prime}=2 a e$
$\therefore e=\frac{S S^{\prime}}{S P+S^{\prime} P}$
$=\frac{\sqrt{386}}{13+25}=\frac{\sqrt{386}}{38}$