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

eq of Tangent is $\frac{\mathrm{x}}{4} \mathrm{cos\theta }+\frac{\mathrm{y}}{9} \mathrm{sin\theta }=1$

$$\begin{gathered} \mathrm{area}=\frac{1}{2\left|{\displaystyle \frac{\mathrm{cos\theta }}{4}} {\displaystyle .\frac{\mathrm{sin\theta }}{9}}\right|} \\ =\frac{36}{\sin 2\mathrm{\theta }} \\ \min imum \mathrm{are}a =\frac{36}{\max \left\{\sin 2\mathrm{\theta }\right\}} \\ =\frac{36}{1}=36 \end{gathered}$$