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Question

The area (in sq units) of the quadrilateral formed by the tangents at the endpoints of the latus recta to the ellipse x29+y25=1 is___ .


A

274

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B

18

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C

272

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D

27

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Solution

The correct option is D

27


Given equation of ellipse is
x29+y25=1a2=9,b2=5a=3,b=5Now,e=1b2a2=159=23Foci=(±ae,0)=(±2,0) and b2a=53


Extermities of one of latusrectum are
(2,53)and(2,53)Equation of tangent at(2,53)isx(2)9+y(53)5=1 or 2x+3y=9
Since the tangent intersects X and Y axes at (92,0) and (0,3) respectively,
Area of quadrilateral =4×Area of ΔPOQ=4×(12×92×3)=27


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