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

The correct option is D $36$

$\frac{x^2}{16}+\frac{y^2}{81}=1$

The equation of a tangent can be written as :

$\frac{x\cos \theta }{a}+\frac{y\sin \theta }{b}=1$

The x intercept $=\frac{a}{\cos \theta }$

The y intercept $=\frac{b}{\sin \theta }$

$\Delta =\frac{1}{2}\times \frac{a}{\cos \theta }\times \frac{b}{\sin \theta }=\frac{ab}{\sin 2\theta }=\frac{36}{\sin 2\theta }$

$\Delta =36$ as $\sin 2\theta$ has maximum value of $1$.