Find the equation of the circle. Find the equation of the circle whose radius is $5$ which touches the circle $x^2 + y^2 - 2x -4y - 20 = 0$ externally at the point $(5,5)$

Express that the circle is through the point $(5,5)$
$$(5-x_c)^2+(5-y_c)^2=5^2,$$
and that the gradients are collinear a this point
$$(x_c-5)(2\cdot5-4)-(y_c-5)(2\cdot 5-2)=0.$$
From the second equation,
$$y_c=\frac{3x_c+5}4,$$
and plugging in the first,
$$x_c^2-10x_c+9=0.$$
The solutions are $(x_c,y_c)=(1,2)$ and $(9,8)$ and we choose the second.
$$\color{green}{(x-9)^2+(y-8)^2=5^5}.$$
