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Question

Area of the quadrilateral formed by tangents at the ends of latus rectum of an ellipse x29+y25=1 is

A
18 sq. units
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B
27 sq. units
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C
92 sq. units
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D
214 sq. units
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Solution

The correct option is A 27 sq. units
Here a2=9,b2=5,b2=a2(1e2)
e2=49e=23
Foci =(±ae,0)=(±2,0)
Ends of latus rectum =(±ae,±b2a)=(±2,±53)
The equation of tangent at one end point of latus rectum (2,53) is
2×x9+53×y5=1
It cuts the co-ordinate axes at (92,0) and (0,3)
x intercept =92, y intercept =3

Area formed by co-ordinate axis and the tangent is same as the area formed by right angled triangle with base 92 and height 3
Area formed by co-ordinate axis and the tangent is
=12×92×3=274
Now, Area of quadrilateral formed by the tangents drawn at the end points of the latum rectums =4×274=27 sq. units.

121720_75965_ans_b7abc49c011549b1afdd55ba93c3819d.png

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