First slide

Ellipse

Question

$Ratio of the greatest and least focal distances of a point on the ellipse 4 x^{2} + 9 y^{2} = 36 is$

Moderate

A

$\frac{7 + 3 \sqrt{5}}{2}$
B

$\frac{5 + 3 \sqrt{6}}{2}$
C

$\frac{7 + 3 \sqrt{5}}{4}$
D

$\frac{7 + 5 \sqrt{5}}{2}$

Solution

$\begin{array}{l} Ellipse \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 (a > b) \\ a^{2} = 9, b^{2} = 4, e = \sqrt{\frac{a^{2} - b^{2}}{a^{2}}} = \sqrt{\frac{9 - 4}{9}} = \frac{\sqrt{5}}{3} \end{array}$
$\begin{aligned} Greatest focal distance = a + a e \\ Least focal distance = a - a e \end{aligned}$
$∴ Required ratio = \frac{a + a e}{a - a e} = \frac{1 + e}{1 - e}$
$\begin{aligned} = \frac{1 + \frac{\sqrt{5}}{3}}{1 - \frac{\sqrt{5}}{3}} \\ = \frac{3 + \sqrt{5}}{3 - \sqrt{5}} \\ = \frac{(3 + \sqrt{5})^{2}}{4} \end{aligned}$
$= \frac{7 + 3 \sqrt{5}}{2}$

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