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Question

The area of the rectangle (in sq. units) formed by the perpendiculars drawn from the centre of the ellipse having major axis and minor axis lengths as 2a and 2b units respectively to the tangent and normal at a point whose eccentric angle is π4 is

A
(a2b2)aba2+b2
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B
(a2+b2)aba2b2
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C
(a2b2)ab(a2+b2)
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D
(a2+b2)(a2b2)ab
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Solution

The correct option is A (a2b2)aba2+b2
Let equation of ellipse be
x2a2+y2b2=1(a>b)
P(θ)=P(π4)
P=(a2,b2)
Tangent at P:xa2+yb2=1
Normal at P: 2ax2by=a2b2
Distance of Normal from O(0,0)
d1=a2b22a2+2b2 units
Distance of tangent from O(0,0)
d2=112a2+12b2
d2=2aba2+b2 units
Area of rectangle =d1d2
=a2b22a2+2b22aba2+b2
=(a2b2)aba2+b2 sq.units

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