The question is:
A circle has the equation ${(x - 3)^2 + (y + 9)^2 = 144}$. A smaller concentric circle has a radius one-third that of the larger one. Find the equation of the smaller circle.
So I would have thought that the radius of the smaller circle would be $\frac {144}{3} = 48$
But the answer to the question is $(x - 3)^2 + (y + 9)^2 = 16$
Which is one third of $\sqrt{144}$, i.e $12$.
Why is it not a third of $144$?
The radius of the first circle is not $144$, but $\sqrt{144}=12$. So, the radius of the smaller circle is $r=12/3=4$.
Notice, the given circle $(x-3)^2+(y+9)^2=144$ or $(x-3)^2+(y+9)^2=(12)^2$ has center at $(3, -9)$ & a radius $12$
Since, the smaller circle is concentric so its center is at $(3, -9)$ & its radius $=\frac{1}{3}\times \text{(radius of larger circle)}=\frac{1}{3}\times (12)=4$
Hence, the equation of smaller circle $$(x-3)^2+(y+9)^2=(4)^2$$ $$\color{red}{(x-3)^2+(y+9)^2=16}$$
