Congratulations to Tony from State College Area High School, Pennsylvania, USA for this solution.

First of all, here is the solution to finding the equation of the orthogonal circle for the circles with centers of $(0,0)$, $(3,0)$, $(9,2)$ and radii respectively of $5$, $4$, and $6$.

As the circles are orthogonal we can draw three right angled triangles. One of the legs of each right triangle is the radius of one of the given circles, the other leg is the radius of the unknown orthogonal circle, and the hypotenuse is the distance between the center of the known circle and the center of the unknown orthogonal circle.

Since we have three given circles, we can use the Pythagorean
Theorem three times to find our answer. Here are the equations that
find the center (h,k) of the orthogonal circle with radius r:

$\begin {eqnarray} 5^2 + r^2 &= h^2 + k^2 \quad (A) \\ 4^2
+ r^2 &= (h-3)^2 + k^2 \quad (B) \\ 6^2 + r^2 &= (h-9)^2 +
(k-2)^2 \quad (C) \\ \end {eqnarray}$

Now subtract equation (B) from equation (A) to get $18 = 6h$,
so $h=3$. Now plug $h=3$ in equation (A) to get $r^2 + 16 = k^2.$
Next plug $h=3$ and $k^2 = r^2 + 16$ into equation (C) to get
:

\[r^2 + 36 = (-6)^2 + (r^2 + 16) -4k + 4\]

from which we find $k = 5.$ Plug $k=5$ in equation (A) to get
$r=3.$ So now we have all we need to make the equation of the
orthogonal circle: $h=3$, $k=5$ and $r=3$, so the equation of the
orthogonal circle is $(x-3)^2 + (y-5)^2=9.$

When we try to find the orthogonal circle for the second
example, we end up with simultaneous equations which are fruitless,
unsolvable. This means the second equation has an orthogonal line,
in this case a fairly obviously one with $y=x.$

Now I will prove that you have an orthogonal line when the
centers of the circles are collinear, otherwise you have an
orthogonal circle.

Any orthogonal line would have to form a right angle with
every given circle. When a line forms a right angle with the
circumference of the circle, that means it passes through the
center of the circle, so our orthogonal line passes through the
centers of all the circles. The only way this is possible for our
line to accomplish this (since two points determine a line, three
points determine a plane), is if all three centers are
collinear.

Therefore, lines can handle every case of collinear centers,
and since an orthogonal line or circle can always be drawn, every
other case is covered by an orthogonal circle.

[We have used Tony's American spelling of center here.
Ed.]