# 2013 AIME II Problems/Problem 10

Given a circle of radius , let be a point at a distance from the center of the circle. Let be the point on the circle nearest to point . A line passing through the point intersects the circle at points and . The maximum possible area for can be written in the form , where , , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .

## Solution

Now we put the figure in the Cartesian plane, let the center of the circle , then , and

The equation for Circle O is , and let the slope of the line be , then the equation for line is

Then we get , according to Vieta's formulas, we get

, and

So,

Also, the distance between and is

So the ares $S=0.5ah=\frac{-4k\sqrt{(16-8\sqrt{13})k^2-13}}{k^2+1}

Then the maximum value of$ (Error compiling LaTeX. ! Missing $ inserted.)S\frac{104-26\sqrt{13}}{3}104+26+13+3=\boxed{146}$