The equation of the circle and its chord are respectively $x^2+y^2=a^2$ and $x\cos \alpha +y\sin \alpha =p$ .The equation of the circle of which this chord is a diameter, is 

The equation of the required circle is

$x^2+y^2-a^2+\lambda (x\cos \alpha +y\sin \alpha -p)=0$         (i)

The centre of this circle is $\left(-\frac{\lambda }{2}\cos \alpha ,-\frac{\lambda }{2}\sin \alpha \right)$ which lies on  

$\begin{array}{l}x\cos \alpha +y\sin \alpha =p.\\ \therefore \lambda =-2p\end{array}$

Putting the value of Ain (i), the equation of the required circle is 

$x^2+y^2-a^2-2px\cos \alpha -2py\sin \alpha +2p^2=0$