Find the equation of pair of tangents from (1,3) to the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}+4\mathrm{y}-11=0$ and also find angle between them.

Equation of the circle is
$\begin{array}{l}\mathrm{S}={\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}+4\mathrm{y}-11=0\\ \mathrm{r}=\sqrt{1+4+11}=4\\ {\mathrm{S}}_{11}=1+9-2+12-11=9\\ \tan \mathrm{\theta }=\frac{\mathrm{r}}{\sqrt{{\mathrm{S}}_{11}}}=\frac{4}{3}\\ \cos 2\mathrm{\theta }=\frac{1-{\tan }^2\mathrm{\theta }}{1+{\tan }^2\mathrm{\theta }}=\frac{1-\frac{16}{9}}{1+\frac{16}{9}}=\frac{-7}{25}\\ 2\mathrm{\theta }={\cos }^{-1}\left(\frac{7}{25}\right)\end{array}$
angle between the tangents $={\cos }^{-1}\left(\frac{7}{25}\right)$