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
B
a2+b22 sq. units
C
(a+b)22 sq. units
D
a2+ab+b23sq. units

Solution

## The correct option is A $$ab$$ sq. unitsThe 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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