I am supposed to find two pairs of points the circle meeting the following constraints passes through.
A circle cutting $x^2+y^2=4$ orthogonally and having its center on the line $2x-2y+9=0$ passes through two fixed points. Find them.
My Attempt
Let the circle be $x^2+y^2+2gx+2fy+c=0$ where $g,f,c\in \mathbb{R}$. Since the circle cuts $x^2+y^2=4$ orthogonally. So by using the condition for orthogonality we get that $c=4$. Now using the fact that the centre of the circle lies on $2x-2y+9=0$, we have $-2g+2f+9=0$. So the equation of the circle can be written as $x^2+y^2+2gx+(2g-9)y+4=0$.
I do not know how to proceed. Any hints would be appreciated. Kindly also inform about the techniques for finding fixed points for any curve. Thanks