Two circles with equal radii are externally tangent
at a point on the parabola y = x². One of the circles
is also tangent to the x-axis while the other is also
tangent to the y-axis. Find the radius of both circles.

At a given point on the parabola, the slope is 2*x. The centers of the circles will lie on a straight line perpendicular to the parabola at that point, so its equation would be y = -1/(2*x0) * (x - x0) + y0, using (x0,y0) to represent the point on the parabola.

Considering the first quadrant:

Let deltax be the difference in the x-coordinate of the circle touching the x-axis and x0. The x-coordinate of the circle touching the y-axis is x0 - deltax.  That x-coordinate, being the radius of each circle is also equal to the y-coordinate of that first-mentioned circle, so we can substitute and find:

x0 - deltax = -1/(2*x0) * deltax + x0^2

We also need to take into consideration that the distance along the perpendicular line from the point on the parabola to the center of each circle is also equal to the radius of the circle, or x0 - deltax, which gives a second equation for the solution, but what I get seems to be a fourth degree equation. Perhaps others could simplify and solve.

Posted by Charlie on 2013-07-13 15:46:14