A tangent drawn to the ellipse $\frac{{\mathrm{x}}^2}{25}+\frac{{\mathrm{y}}^2}{16}=1$ so that the part intercepted by the axes is minimum. The length of this part of the tangent is

 $\frac{{\mathrm{x}}^2}{25}+\frac{{\mathrm{y}}^2}{16}=1$
eqn of tangent is

$$\begin{gathered} \frac{\mathrm{x}}{5}\cos \mathrm{\theta }+\frac{\mathrm{y}}{4}\sin \mathrm{\theta }=1 \\ \frac{\mathrm{x}}{5 \sec \mathrm{\theta }}+\frac{\mathrm{y}}{4 \mathrm{cosec} \mathrm{\theta }}=1 \end{gathered}$$

its meet coordinate axes A, B

$$\begin{gathered} \mathrm{A}(5\sec \mathrm{\theta }, 0) \mathrm{B}(0, 4\mathrm{cosec} \mathrm{\theta }) \\ {\mathrm{AB}}^2=25{\sec }^2\mathrm{\theta }+16{\mathrm{cosec}}^2\mathrm{\theta } \\ =25+25{\tan }^2\mathrm{\theta }+16+16{\cot }^2\mathrm{\theta } \\ \mathrm{A}.\mathrm{M}\ge \mathrm{G}.\mathrm{M} \\ \frac{25{\mathrm{Tan}}^2\mathrm{\theta }+16{\cot }^2\mathrm{\theta }}{2} \\ \ge \sqrt{25{\mathrm{Tan}}^2\mathrm{\theta }.16{\cot }^2\mathrm{\theta }} \\ 25{\mathrm{Tan}}^2\mathrm{\theta }+16{\cot }^2\mathrm{\theta } \\ \ge 2\times 5\times 4 \\ \ge 40 \\ {\mathrm{AB}}^2\ge 41+40 \\ \mathrm{AB}\ge 9 \end{gathered}$$