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

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

$\frac{\sqrt{386}}{25}$

$\frac{\sqrt{386}}{20}$

$\frac{\sqrt{386}}{45}$

answer is A.

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Detailed Solution

$S = (5, 12) S^{1} = (24, 7)$

$P (0, 0)$

$S P = 13 S^{1} P = 25 S S^{1} = \sqrt{386}$

$S P + S^{1} P = 2 a S S^{1} = 2 a e$

$e = \frac{S S^{1}}{S P + S^{1} P} = \frac{\sqrt{386}}{38}$

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