10-2. Parabola, Ellipse, Hyperbola
hard

If the ellipse $\frac{ x ^{2}}{ a ^{2}}+\frac{ y ^{2}}{ b ^{2}}=1$ meets the line $\frac{x}{7}+\frac{y}{2 \sqrt{6}}=1$ on the $x$-axis and the line $\frac{x}{7}-\frac{y}{2 \sqrt{6}}=1$ on the $y$-axis, then the eccentricity of the ellipse is

A

$\frac{5}{7}$

B

$\frac{2 \sqrt{6}}{7}$

C

$\frac{3}{7}$

D

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

(JEE MAIN-2022)

Solution

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ meets the line $\frac{x}{7}+\frac{y}{2 \sqrt{6}}=1$ on the $x$-axis

So, $a=7$

and $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ meets the line $\frac{x}{7}-\frac{y}{2 \sqrt{6}}=1$ on the $y$-axis

So, $b=2 \sqrt{6}$

Therefore, $e^{2}=1-\frac{b^{2}}{a^{2}}=1-\frac{24}{49}$

$e=\frac{5}{7}$

Standard 11
Mathematics

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