The acute angle between the pair of tangents drawn to the ellipse $2{\mathrm{x}}^2+3{\mathrm{y}}^2=5$ from the point (1, 3) is

Equation of tangent to the ellipse $2{\mathrm{x}}^2+3{\mathrm{y}}^2=5$ is $\mathrm{y}=\mathrm{mx}\pm \sqrt{\frac{5}{2}{\mathrm{m}}^2+\frac{5}{3}}$

It pass through (1, 3)

$\begin{array}{l}3=\mathrm{m}\pm \sqrt{\frac{5}{2}{\mathrm{m}}^2+\frac{5}{3}}\\ \Rightarrow {\left(3-m\right)}^2=\frac{15m^2+10}{6}\\ \Rightarrow 6\left(9+m^2-6m\right)=15m^2+10\Rightarrow 9m^2+36m-44=0\\ \\ 3{\mathrm{m}}^2+12\mathrm{m}-\frac{44}{3}=0\end{array}$

Let $\mathrm{\theta }$ be the angle between the tangents

$\begin{array}{l}\tan \mathrm{\theta }=\left|\frac{{\mathrm{m}}_1-{\mathrm{m}}_2}{1+{\mathrm{m}}_1{\mathrm{m}}_2}\right|\\ \tan \mathrm{\theta }=\left|\frac{3\sqrt{320}}{-35}\right|\\ \mathrm{\theta }={\tan }^{-1}\left(\frac{24}{7\sqrt{5}}\right)\end{array}$