Mock Geometry AIME 2011 Problems/Problem 10
Circle is defined by the equation where is a positive real number. Circle passes through the center of and its center lies on the line Suppose that one of the tangent lines from the origin to circles and meets and at respectively, that where is the origin, and that the radius of is . What is ?
Let the centers of be and let their radii be respectively. From the given information, and for some . Using the distance formula between and yields . From right triangle , we have . Rearranging yields .
Similarly, using the distance formula between and yields . From right triangle , we have . Substituting for and rearranging yields .
From the distance formula, . Setting the previous two equations equal to each other yields . The terms nicely cancel out, leaving . Thus .