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.

(In reply to a start by Charlie)

DEFDBL A-Z
CLS
FOR x0 = .8201941# TO .8201941# STEP .0000001#
  y0 = x0 * x0
  b = -1 / (2 * x0)

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

  newy = y0 + b * deltax
  leftx = x0 - deltax
  deltay = y0 - newy
  dist = SQR(deltay ^ 2 + deltax ^ 2)
  PRINT USING "###.#######"; x0; y0; leftx; newy; dist; newy - dist
  PRINT x0 + deltax, y0 + deltay
NEXT

finds

 0.8201941  0.6727184  0.4424272  0.4424272  0.4424272 -0.0000000
1.197960993479501           .903009516829121

The point on the parabola is (0.8201941, 0.6727184).
The radius of each circle is 0.4424272.
The left-hand circle is centered at (0.4424272, 0.9030095).
The right-hand circle is centered at (1.197961, 0.4424272).

Posted by Charlie on 2013-07-13 17:07:42