Angle between the tangents drawn from the point (5,4) to the ellipse $\frac{{\mathrm{x}}^2}{25}+\frac{{\mathrm{y}}^2}{16}=1$ is 

$\begin{array}{l}\tan \mathrm{\theta }=\frac{2\mathrm{ab}\sqrt{{\mathrm{S}}_{11}}}{{\mathrm{x}}_1^2+{\mathrm{y}}_1^2-\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)}\\ \Rightarrow \tan \mathrm{\theta }=\mathrm{\infty }\Rightarrow \mathrm{\theta }={90}^{\circ }\\ OR\end{array}$

(5,4) lies on the director circle$x^2+y^2=41$ of the ellipse$\frac{{\mathrm{x}}^2}{25}+\frac{{\mathrm{y}}^2}{16}=1$

$$\begin{gathered} \therefore \tan gents are perpendicular \\ \Rightarrow angle={90}^{\circ } \end{gathered}$$