What is the shortest distance from the origin to the circle?

I think the easiest way to do this is to parameterize the circle via a single variable and solve the minimization problem.

$\text{let }p \text{ be a point on the circle given by }(x-12)^2 + (y+5)^2 = 9 \\ \text{then p can be described as }\\ p=(3 \cos(\theta)+12,~3\sin(\theta)-5)$

$\text{The squared distance of this point to the origin is }\\ d^2 = (3 \cos(\theta)+12)^2 + (3\sin(\theta)-5)^2 =\\ 9 \sin ^2(\theta )-30 \sin (\theta )+9 \cos ^2(\theta )+72 \cos (\theta )+169 = \\ -30 \sin (\theta )+72 \cos (\theta )+178 = \\ 78 \cos(\theta+\arctan(5/12))+178$

$d^2 \text{will clearly be minimized when }\\ \theta + \arctan(5/12) = \pi\\ \theta = \pi - \arctan(5/12)\\ \text{at this angle }d^2 = -78 + 178 = 100\\ d=10$