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)

$\frac{\sqrt{386}}{38}$

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

$P(0,0)$ is a point on the conic.

$SP=\sqrt{25+144}=13$

$S^{\prime }P=\sqrt{576+49}=\sqrt{625}=25$

$S{S}^{\prime }=\sqrt{{(24-5)}^2+{(7-12)}^2}=\sqrt{{19}^2+5^2}=\sqrt{386}$

If the conic is an ellipse, then

$SP+S^{\prime }P=2a$ and $S{S}^{\prime }=2ae$

$\therefore e=\frac{S{S}^{\prime }}{SP+S^{\prime }P}=\frac{\sqrt{386}}{13+25}=\frac{\sqrt{386}}{38}$