Question

Minimum area of the triangle by any tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with the coordinate axes is

• A. $\frac{a^2+b^2}{2}$
• B. $\frac{{(a+b)}^2}{2}$
• C. $ab$
• D. $\frac{{(a-b)}^2}{2}$

Answer 1

Answer: (C)

$ab$

Equation of tangent at $(a\cos \theta ,b\sin \theta )$ is
$\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$
then, $P=(\frac{a}{\cos \theta },0);Q=(0,\frac{b}{\sin \theta })$
Area of $OPQ=\frac{1}{2}\left|\left(\frac{a}{\cos \theta }\right)\left(\frac{b}{\sin \theta }\right)\right|=\frac{ab}{\left|{\sin 2\theta }^{}\right|}$
$\therefore$ Area $=ab$