Family of Circles

A circle cutting the circle $x^2+y^2=4$ orthogonally and having its centre on the line $2x-2y+9=0,$ passes through two fixed points. These points are

Equation of a circle having radius equal to twice the radius of the circle $x^2+y^2+\left(2p+3\right)x+\left(3-2p\right)y+p-3=0$ and touching it at the origin is