Area (in sq, units) of quadrilateral formed by tangents at end points of latera recta to ellipse fra

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10-2. Parabola, Ellipse, Hyperbola

hard

The area (in sq, units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{5} = 1$ is :

A

$27$

B

$\frac{{27}}{4}$

C

$18$

D

$\frac{{27}}{2}$

(JEE MAIN-2015)

Solution

$a=3, b=\sqrt{5}$

$e=\sqrt{1-\frac{5}{9}}=\frac{2}{3}$

foci $=(\pm 2,0)$

tangent at $P \Rightarrow \frac{2 x}{9}+\frac{5 y}{3.5}=1$

$\frac{2 x}{9}+\frac{y}{3}=1$

$2 x+3 y=9$

Area of quadrilateral

$=4 \times \text { (area of triangle } \mathrm{QCR})$

$=\left(\frac{1}{2} \times \frac{9}{2} \times 3\right) \times 4=27$

Standard 11

Mathematics

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