If ellipse frac{ x ^{2}}{ a ^{2}}+frac{ y ^{2}}{ b ^{2}}=1 meets line frac{x}{7}+frac{y}{2 √{6}}=1

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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

$\frac{5}{7}$

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

$\frac{3}{7}$

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

(JEE MAIN-2022)

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

hard

(IIT-2009)

normal

normal

hard

(JEE MAIN-2014)

normal