First slide

Ellipse

Question

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

Moderate

A

$\frac{\sqrt{386}}{12}$
B

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

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

$\sqrt{2}$

Solution

$The two foci are S (5, 12) and S^{'} (24, 7) and the ellipse$
$passes through origin O, then$
$\begin{array}{l} S O = \sqrt{25 + 144} = 13; S^{'} O = \sqrt{576 + 49} = 25 \\ and S S^{'} = \sqrt{386} \end{array}$
$\begin{array}{l} If a conic be an ellipse, then S O + S^{'} O = 2 a and S S^{'} = 2 a e \\ ∴ e = \frac{S S^{'}}{S^{'} O + S O} = \frac{\sqrt{386}}{38} \end{array}$

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