<s>[INST] Point of Tangency between Two Circles I made sure to look around this site a couple times before asking my question. A lot of people have already asked how to find the point of tangency or intersection between two circles, but none of them could really help me solve the problem.
I'm being asked to find the point of tangency between two circles and all I am given are just the two circles -- no equations of tangent lines, etc. Here's a picture of what I'm working with:

The equation of the green circle is $x^2$ + $y^2$ = 5.
The equation of the orange circle was given as $x^2$ - 12x + $y^2$ + 6y = -25, which I rewrote as ($x-6)^2$ + $(y+3)^2$ = 20.
How do I find the coordinates of the circles' point of tangency?
Thank you in advance!
 [/INST] The equations are:
\begin{gather}
    x^2 + y^2 = 5 \tag{1} \\
    x^2 -12x + y^2 + 6y + 25 = 0 \tag{2}
\end{gather}
If you substitute  (1) into (2), you get
\begin{align}
    -12 x + 6y + 30 &= 0 \implies y=2x-5\tag{3}
\end{align}
Substitute this back into (1):
\begin{align}
    x^2 + (2x-5)^2 &= 5 \\\implies
    x^2 + 4x^2 - 20x + 25 &= 5 \\\implies
    5x^2 - 20x + 20 &= 0 \\\implies
    x^2 -4 x + 4 &= 0 \\\implies
    (x-2)^2 &= 0
\end{align}
Therefore $x=2$.  By (3), $y=-1$.
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