<s>[INST] 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 $(5,5)$
 [/INST] The circle: $x^2+y^2-2x-4y-20=0$ has center $(1, 2)$ & a radius $=\sqrt{(-1)^2+(-2)^2-(-20)}=5$ & the unknown circle has a radius $5$ Hence the point $(5, 5)$ is the mid point of line joining their centers
Let the center of unknown circle be $(a, b)$ then the point $(5, 5)$ is mid point of lines joining the centers $(a, b)$ & $(1, 2)$ hence we have $$\left(\frac{a+1}{2}, \frac{b+2}{2}\right)\equiv(5, 5)$$ by comparing the corresponding coordinates we get $$\frac{a+1}{2}=5\implies a=9$$ $$\frac{b+2}{2}=5\implies b=8$$ Hence the equation of the circle having center $(9, 8)$ & a radius $5$ is given as $$(x-9)^2+(y-8)^2=5^2=25$$
$$\color{blue}{(x-9)^2+(y-8)^2=25}$$
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