jmc

We complete the square by observing that the equation for the circle is equivalent to $$(x^2-24x+144)+(y^2+10y+25)-9=0,$$ which is also equivalent to $$(x-12{)}^2+(y+5{)}^2=3^2.$$ Hence the center of the circle is $(12,-5)$ and by the Pythagorean theorem, the distance from the origin to the center of the circle is $13$ (we can also recall that we have a $5-12-13$ triangle). Since the radius of the circle is $3$, the shortest distance from the origin to the circle is the difference of the distance from the center of the circle to the origin less the radius which is $13-3=\boxed{10}$.