Question

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

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

Answer 1

Answer: (C)

Equation of tangent at $(a\cos \theta ,b\sin \theta )$ to the ellipse is
$\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$

Coordinates of $P$ and $Q$ are
$\left(\frac{a}{\cos \theta },0\right)$ and $\left(0,\frac{b}{\sin \theta }\right)$, respectively.
Now, area of
$\Delta OPQ=\frac{1}{2}\left|\frac{a}{\cos \theta }\times \frac{b}{\sin \theta }\right|=\frac{ab}{|\sin 2\theta |}$
$\therefore$ Minimum area $=ab$