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
$\Delta O P Q=\frac{1}{2}\left|\frac{a}{\cos \theta} \times \frac{b}{\sin \theta}\right|=\frac{a b}{|\sin 2 \theta|}$
$\therefore $ Minimum area $=a b$