Here's yet another way to do that:

is the square of the distance from (0, 0) to (x, y) and will be minimized when the distance is

..?.. and the shortest distance is always on a straight line. Draw a Straight line from (0, 0) to (-5, 12), center of the circle. The point on the circle closest to (0, 0) will be the point where that line crosses the circle. Find the equation of that line (since it passes through (0, 0) it can be written as y= mx) and solve that and the equation of the circle. That will result in a quadratic which has two solutions- one solution will be point that minimizes

, the other the point where it is maximized.