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Circles & Parabola (Posted on 2013-07-13) Difficulty: 4 of 5

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.

  Submitted by Bractals    
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Solution: (Hide)

Let (a,r) and (r,b) be the centers of the circles tangent
to the x-axis and y-axis respectively (where r is the
radius desired). Let (c,c2) be the point of tangency on
the parabola (WOLOG the first quadrant branch will
be used). The centers of the circles must lie on the
normal line at the point (c,c2) (see my reply to Charlie's
post for a correction to the problem):
    c2-r     c2-b     -1
   ------ = ------ = ----                     (1)
    c-a      c-r      2c
The distance from the centers to the point (c,c2) is r:
   (c2-r)2 + (c-a)2 = (c2-b)2 + (c-r)2 = r2    (2)
Combining (1) and (2) to eliminate the coordinates
a and b to give:
   (c2-r)2 + 4c2(c2-r) = r2                    (3) 

                    and

   (c-r)2/4c2 + (c-r)2 = r2                    (4)
Which in turn give
   r = wc2/(w+1)                              (5)

                and

   r = wc/(w+2c)                              (6)

                where w2 = 4c2+1.
Combine (5) and (6) to find c:
   wc2/(w+1) = wc/(w+2c)  

   ⇒  8c2-9c+2 = 0

   ⇒  c = [9+s√(17)]/16                       (7)

                where s = ±1.
The point (c,c2) is the midpoint of the line segment
joining the centers of the circles. Therefore,
   c = (a+r)/2                                (8)
Combining (1) and (8) gives
    c2-r      1
   ------ = ----                              (9)
    c-r      2c
Therefore,
        c(2c2-1)
   r = ----------                             (10)
          2c-1
Combining (7) and (10) gives
   r = 3[23-s√(17)]/128                       (11)

QED

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionSolutionHarry2013-07-14 06:43:52
Solutioncomputer approximationCharlie2013-07-13 17:07:42
Hints/Tipsa startCharlie2013-07-13 15:46:14
re: thoughts - Problem CorrectionBractals2013-07-13 14:30:53
Some ThoughtsthoughtsCharlie2013-07-13 13:55:05
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