Question

If $(5,12)$ and $(24,7)$ are the foci of a conic passing through the origin, then the eccentricity of conic can be:

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

Answer 1

Answer: (A), (B)

The correct options are
A $\frac{\sqrt{386}}{12}$
B $\frac{\sqrt{386}}{38}$

Let $S(5,12)$ and $S^{\prime }(24,7)$ be the two foci of the conic.
It passes through $P(0,0)$
$\Rightarrow SP=\sqrt{5^2+{12}^2}=13$
$\Rightarrow S^{\prime }P=\sqrt{{24}^2+7^2}=25$
$\Rightarrow S{S}^{\prime }=\sqrt{{19}^2+5^2}=\sqrt{386}$

If the conic is ellipse, then
$SP+S^{\prime }P=2a$ and $S{S}^{\prime }=2ae$
$\therefore e=\frac{S{S}^{\prime }}{SP+S^{\prime }P}=\frac{\sqrt{386}}{38}$

If the conic is hyperbola, then
$|SP-S^{\prime }P|=2a$ and $S{S}^{\prime }=2ae$
$\therefore e=\frac{\sqrt{386}}{12}$