Locus of the point the chord of contact of which subtends an angle 2$\theta$ at the centre of
circle $x^2+y^2=r^2$ is 
 

$$\begin{gathered} Let P(x_1,y_1) be the point from where \tan gents are drawn to the chord \\ \Rightarrow S_{11}= {x_1}^2+{y_1}^2-r^{2}and radius =r \\ Angle between the \tan gents=\tan \frac{\theta }{2}=\frac{r}{\sqrt{S_{11}}} \\ given angle between the \tan gents=2\theta \\ \Rightarrow \tan \left(\theta \right)=\frac{r}{\sqrt{{x_1}^2+{y_1}^2-r^2}} \\ \Rightarrow locus is x^2+y^2-r^2=r^{2}co{t}^2\theta \\ \Rightarrow x^2+y^2=r^2\cos e{c}^2\theta \end{gathered}$$