Answer
$8$
Work Step by Step
Step 1. Rewrite the given equation as $x^2-10x+25+y^2-24y+144=25+144-144$ or $(x-5)^2+(y-12)^2=25$ which gives a circle with center at $(5,12)$ and radius $r=5$
Step 2. The shortest distance from the origin to the circle can be found by connecting a line from the origin to the center of the circle and find the difference of the center to origin distance and the radius.
Step 3. The distance from the origin to the center of the circle is $d=\sqrt {12^2+5^2}=13$, thus the shortest distance is $13-5=8$
