Equation of a circle subtending a given angle

Question

The question is to find out the equation of the following circle

For case like $\theta=90°$ the equation is simply $(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$.But I couldn't generalise this Any ideas?Thanks.

Answer 1

Hint:

If $A=(x_1,y_1)$ , $B=(x_2,y_2)$ and $l=\overline{AB}=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$ is the lenght of the chord, than the radius of the searched circle is $$ r=\frac{l}{2\sin \theta} $$

and the center of the circle stay on a point $C=(x,y)$ such that: $$ \begin{cases} (x-x_1)^2+(y-y_1)^2=r^2\\ (x-x_2)^2+(y-y_2)^2=r^2 \end{cases} $$

You can solve this system (that represents the intersection of two circles)?