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Question

The minimum area of triangle formed by the tangent to the ellipse $$\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ 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+b23sq. units
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Solution

The correct option is A $$ab$$ sq. units
The equation of tangent at $$(a\cos\theta ,b\sin\theta )$$ is, 

$$\displaystyle \frac{x\,
\cos\theta }{a}+\frac{y\: \sin\theta}{b }=1$$ 

It meets the coordinate axes at $$A  (0,b \cos ec\, \theta ),B=(a\sec\, \theta,0 )$$

Thus area of triangle is $$=\displaystyle \frac{ab}{2\sin\, \theta \cos\theta }=\frac{ab}{\sin2\theta }\geq ab.$$

Mathematics

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