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 xcoordinate of the circle touching the xaxis and x0. The xcoordinate of the circle touching the yaxis is x0  deltax. That xcoordinate, being the radius of each circle is also equal to the ycoordinate of that firstmentioned 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 20130713 15:46:14 