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.

Let the point of contact be T, (t, t²), and radii r.

Pythagoras gives:                      r² = (t² – r)² + (t – r)²     (1)

Gradient of normal at T gives:     1/(2t) =(t² – r)/(t – r)     (2)

(2) implies:                                r = t(2t² – 1)/(2t – 1)      (3)

which, when substituted into (1) gives 8t² – 9t + 2 = 0 (eventually!):

Thus:                                        t = (9 +/- sqrt 17)/16
and, from (3), (eventually!):       r = 3(23 -/+ sqrt 17)/128

One root seems to agree with Charlie’s numerical results, giving:

r = 3(23 - sqrt 17)/128 ~ 0.442427
T = ((9 + sqrt 17)/16, (49 + 9 sqrt 17)/128) ~ (0.820194, 0.672718)
Upper circle: centre at (0.442427, 0.90301) and touching the y axis
Lower circle: centre at (1.19796, 0.442427) and touching the x axis

and the other root also gives a valid solution:

r = 3(23 + sqrt 17)/128 ~ 0.635698
T = ((9 – sqrt 17)/16, (49 - 9 sqrt 17)/128) ~ (0.304806, 0.0929066)
Upper circle: centre at (-0.026086, 0.635698) and touching the x axis
Lower circle: centre at (0.635698, -0.449885) and touching the y axis

Posted by Harry on 2013-07-14 06:43:52