Let frac{x^2}{a^2}+frac{y^2}{b^2}=1, a>b be an ellipse with foci F₁ and F₂ . Let AO be its se

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10-2. Parabola, Ellipse, Hyperbola

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Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$ be an ellipse with foci $F_1$ and $F_2$. Let $AO$ be its semi-minor axis, where $O$ is the centre of the ellipse. The lines $A F_1$ and $A F_2$, when extended, cut the ellipse again at points $B$ and $C$ respectively. Suppose that the $\triangle A B C$ is equilateral. Then, the eccentricity of the ellipse is

A

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

B

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

C

$\frac{1}{3}$

D

$\frac{1}{2}$

(KVPY-2018)

Solution

(d)

Given, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

$A B C$ is an equilateral traingles.

$\ln \Delta A O F_2^1$

$\tan 30^{\circ}=\frac{O F_2}{O A}$

$\frac{1}{\sqrt{3}} =\frac{a e}{b}$

$b =\sqrt{3} a e$

$b^2 =3 a^2 e^2$

$\frac{b^2}{a^2} =3 e^2$

$4 e^2 =1$

$e^2 =\frac{1}{4} \Rightarrow e=\frac{1}{2}$

Standard 11

Mathematics

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