The point you’re looking for is the intersection of g(x, a) = \frac{x^2}{2a} - \frac{a}{2} and h(x,b,c) = \frac{(x-c)^2}{2b} - \frac{b}{2} (using the notation of your Desmos page), right? [Yay, LaTeX works!]
To find the x-coordinate of the intersection point as a function of a,b,c, you can set g=h and solve:
\frac{x^2}{2a} - \frac{a}{2} = \frac{(x-c)^2}{2b} - \frac{b}{2}
This is equivalent to the following quadratic equation:
\frac{b-a}{2ab} x^2 + \frac{c}{b}x +\left(\frac{b-a}{2}-\frac{c^2}{2b}\right) = 0
whose solutions are:
x = \frac{ac}{b-a}\left(-1 \pm \sqrt{1 - \frac{1}{ac^2}(b-a)(b^2-ab-c^2)}\right)
From fooling around on Desmos it looks like the solution with the + sign (purple vertical line) is the correct one:
To get the y-coordinate you can plug this expression for x into either g or h. I haven’t done the algebra to simplify it, but I added the line (purple horizontal line) and the final circle (purple) to my Desmos link above.