Q:

How to determine family of circles passing through two given points?

The question asks to show that the equation of any circle passing through two given points takes a certain form. I have obtained the points as being $(2,1)$ and $(2,-1)$ but I'm not sure as to how to determine the general form of the family of circles that pass through these points from just this information.
For anyone that's interested, I've posted the entire question below.

Show that the equation of any circle passing through the points of intersection of the ellipse
$$(x+2)^2+2y^2=18$$
and the ellipse
$$9(x-1)^2+16y^2=25$$
can be written in the form
$$x^2-2ax+y^2=5-4a.$$

Thanks guys.

A:

For the first one,  if the center is $(h,k)$
We have $(2-h)^2+(1-k)^2=(2-h)^2+(1+k)^2$ (each being the radius$^2$)
$\implies4k=0\iff k=0$
So, the equation will be $(x-h)^2+(y-0)^2=(2-h)^2+1$

The equation any  two dimensional curve passing through the intersection of the given ellipses can be written as  
$$(x+2)^2+2y^2-18+k[9(x-1)^2+16y^2-25]=0$$ where $k$ is an arbitrary constant
For a circle, the coefficients of $x^2,y^2$ must be same
Use this fact to determine the value of $k$

