Question

If $(5,12)$ and $(24,7)$ are the focii of a conic passing through the origin, then the eccentricity of 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: (A), (D)

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

Let the foci be $S(5,12)$ and $S^{\prime }(24,7)$
The conic passes through the origin.
$\therefore$ Let $P(0,0)$
$\therefore S^{\prime }P-SP=12$ and $S^{\prime }P+SP=38$
Distance between focii $=2ae$
$\sqrt{386}=2ae$
If conic is ellipse then $S^{\prime }P+SP=38=2a$
$\therefore$ Eccentricity $e=\frac{\sqrt{386}}{38}$
And if conic is hyperbola then $S^{\prime }P-SP=12=2a$
$\therefore$ Eccentricity $e=\frac{\sqrt{386}}{12}$