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)

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 $\sin 2\theta =1$