Let frac{x^2}{a^2}+frac{y^2}{b^2}=1(b < a) , be a ellipse with major axis A B and minor axis C D

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10-2. Parabola, Ellipse, Hyperbola

hard

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(b < a)$, be a ellipse with major axis $A B$ and minor axis $C D$. Let $F_1$ and $F_2$ be its two foci, with $A, F_1, F_2, B$ in that order on the segment $A B$. Suppose $\angle F_1 C B=90^{\circ}$. The eccentricity of the ellipse is

A

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

B

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

C

$\frac{\sqrt{5}-1}{2}$

D

$\frac{1}{\sqrt{5}}$

(KVPY-2020)

Solution

(c)

Here, $\angle F_1 C B=90^{\circ}$

$\frac{-b}{a} \cdot \frac{b}{a e}=-1 \Rightarrow \frac{b^2}{a^2}=e \Rightarrow e=1-e^2$

$e^2+e-1=0$

$e=\frac{-1 \pm \sqrt{5}}{2} \stackrel{\overline{5}}{2} \Rightarrow e=\frac{\sqrt{5}-1}{}$

Standard 11

Mathematics

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