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Question

The area of the quadrilateral formed by the tangent at the endpoints of latus recta to the ellipse x29+y25=1 is -

A
274 sq. units
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B
9 sq. units
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C
27 sq. units
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D
272 sq. units
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Solution

The correct option is B 27 sq. units
we have to find the area of the quadrilateral formed by the tangents at the end points of the latus recta of the ellipse x29+y25=1

a=3,b=5
e=1b2a2=159=45=23
Foci =(±ae,0)
=(±3×23,0)=(±2,0)
Ends of latus rectum =(±ae,b2a)
=(±2,±53)
Let the tangent at point (2,53) be xx19+yy15=1
2x9+53×y5=2x9+y3=1
It cuts the coordinates axes at P(92,0) and Q(0,3)
By symmetryarea of quadrilateral =4(12×92×3)
=4×274=27 sq. units

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