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
Then the maximum value of is
So the answer is