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Question

The area of the triangle formed by the tangent at any point of the ellipse x2a2+y2b2=1 with the axes is minimum at the point

A
(a2b2)
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B
(a2,b2)
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C
(a,b)
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D
(a2b2)
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Solution

The correct option is C (a2b2)
Let any point (asinθ,bcosθ)
2xa2+2yy1b2=0
y1=xb2a2y
y1=btanθa
(ybcosθ)=btanθa(xasinθ)
acos2θsinθ+asinθ=x (x intercept)
y=bcosθ+bsin2θcosθ (y intercept)
area=12xy
=1ab2sinθcosθ
=absin2θ
area is minimum when sin2θ=1
So θ=π4

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