Question

The circle $S\equiv x^2+y^2+2gx+2fy+c=0$ subtends an angle $\theta$ at any point $P(x_1,y_1)$ then prove that $\tan \frac{\theta }{2}=\frac{r}{t}=\frac{r}{\sqrt{S^{\prime }}}$ where $t$ is the length of the tangent drawn from $P$ to the circle $S$.

Answer 1

$\angle POQ=\theta$ given $\therefore \angle COP=\theta /2$
$\therefore \tan \frac{\theta }{2}=\frac{r}{t}=\frac{r}{\sqrt{S^{\prime }}}\because t^2=S^{\prime }$.