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Question

The minimum area of a triangle formed by the tangent to the ellipse x2a2+y2b2=1 and coordinate axes is

A
ab sq. units
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B
a2+b22 sq. units
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C
(a+b)22 sq. units
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D
a2+ab+b22 sq. units
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Solution

The correct option is A ab sq. units
Any tangent to the ellipse x2a2+y2b2=1 at P(acosθ,bsinθ) is given by
xcosθa+ysinθb=1


It meets coordinate axes at A(asecθ,0) and B(0,b cosec θ)
Area of OAB=12×|asecθ×b cosec θ|
=absin2θ

For area to be minimum sin2θ should be maximum and we know that maximum value of sin2θ is 1.
max=ab Sq.units

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