The Locus of the point, whose chord of contact w.r.t the circle $x^2+y^2=a^2$ makes an angle $2\alpha$ at the centre of the circle is

Let $P\left(x_1{y}_1\right)$ be a point on the locus Let $PA$ and $PB$ be the tangents drawn from $P$ to the circe $x^2+y^2=a^2$ and the chord $AB$ makes an angle $2\alpha$ at the origin the centre of the circle.
Then $\overline{)AOP}=\alpha ,OA=aand\overline{)OAP}={90}^0$  
From $\Delta OAP,\cos \alpha =\frac{OA}{OP}\Rightarrow \sec \alpha =\frac{OP}{OA}$  
 

$\Rightarrow O{P}^2=O{A}^{2}Se{c}^2\alpha \Rightarrow x_1^2+y_1^2=a^2{\sec }^2\alpha$
Locus is $x^2+y^2=a^2{\sec }^2\alpha$