Question

The minimum area of a triangle formed by any tangent to the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{81}=1$ and the co-ordinate axes is :

• A. 12
• B. 18
• C. 26
• D. 36

Answer 1

Answer: (D)

36

Answer 2

Let $P (4 \cos \theta, 9 \sin \theta)$ be a point on ellipse
equation of tangent $\frac{x}{4} \cos \theta+\frac{y}{9} \sin \theta=1$
Let $A \& B$ are point of intersection of tangent at $P$ with co-ordinate axes.

$A \left(\frac{4}{\cos \theta}, 0\right) B \left(0, \frac{9}{\sin \theta}\right)$
Area of $\Delta OAB =\frac{1}{2}\left(\frac{4}{\cos \theta}\right)\left(\frac{9}{\sin \theta}\right)=\frac{36}{\sin 2 \theta}$
$(\text { Area })_{\min }=36$ as $\sin 2 \theta=1$