An ellipse frac{left(x-x_0right)^2}{a^2}+frac{left(y-y_0right)^2}{b^2}=1 , a > b , is tangent to

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10-2. Parabola, Ellipse, Hyperbola

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An ellipse $\frac{\left(x-x_0\right)^2}{a^2}+\frac{\left(y-y_0\right)^2}{b^2}=1$, $a > b$, is tangent to both $x$ and $y$ axes and is placed in the first quadrant. Let $F_1$ and $F_2$ be two foci of the ellipse and $O$ be the origin with $OF _1 < OF _2$. Suppose the triangle $OF _1 F _2$ is an isosceles triangle with $\angle OF _1 F _2=120^{\circ}$. Then the eccentricity of the ellipse is

A

$\frac{1}{2 \sqrt{3}}$

B

$\frac{2}{3}$

C

$\frac{1}{2}$

D

$\frac{1}{\sqrt{2}}$

(KVPY-2021)

Solution

(c)

$F _1 F _2= OF _1=2 ae$

$OC =\sqrt{ a ^2+ b ^2}$

$\cos 120^{\circ}=\frac{( ae )^2+4( ae )^2- a ^2- b ^2}{2 \cdot ae \cdot 2 ae }$

$\Rightarrow-\frac{1}{2}=\frac{5( ae )^2- a ^2- a ^2\left(1- e ^2\right)}{4 a ^2 e ^2}$

$\Rightarrow 8 e ^2=2$

$e =\frac{1}{2}$

Standard 11

Mathematics

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