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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.

See The Solution Submitted by Bractals    
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Some Thoughts thoughts | Comment 1 of 5

As written, the puzzle does not require that the point of external tangency of the two circles also be a point of tangency of the circles with the parabola, only that the point is on the parabola.

DEFDBL A-Z
CLS
PRINT "b\c "; :
FOR c = .5 TO 2 STEP .1: PRINT USING "##.#"; c; : NEXT: PRINT
FOR b = .32 TO .66 STEP .01
PRINT USING "#.##"; b;
FOR c = .5 TO 2 STEP .1
  amb = 4 * b * b - (c - b) * (c - b)
  IF amb < 0 THEN
    PRINT "iiii";
  ELSE
    amb = SQR(amb)
    a = amb + b
    v = ((a + b) / 2) ^ 2 - (c + b) / 2
    PRINT USING "##.#"; v;
  END IF
NEXT: PRINT
NEXT: PRINT

DO: LOOP UNTIL INKEY$ > ""

 

The above program finds the centers (a,b) and (b,c) of the two circles of radius b such that they are separated by a distance of 2*b and therefore are externally tangent, and as the y and x coordinates respectively are b units from the axes they satisfy the other requirement. What's left is to show for each of these the difference between y and x^2, which is tabulated below:


b\c  0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
0.32-0.0-0.1-0.2-0.3-0.4iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.33 0.0-0.1-0.2-0.2-0.4iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.34 0.0-0.0-0.1-0.2-0.3-0.5iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.35 0.1-0.0-0.1-0.2-0.3-0.4iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.36 0.1 0.0-0.1-0.2-0.3-0.4iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.37 0.1 0.0-0.0-0.1-0.2-0.4-0.5iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.38 0.1 0.1-0.0-0.1-0.2-0.3-0.5iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.39 0.2 0.1 0.0-0.1-0.2-0.3-0.4iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.40 0.2 0.1 0.0-0.0-0.1-0.3-0.4iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.41 0.2 0.1 0.1-0.0-0.1-0.2-0.4-0.5iiiiiiiiiiiiiiiiiiiiiiiiiiii
0.42 0.2 0.2 0.1 0.0-0.1-0.2-0.3-0.5iiiiiiiiiiiiiiiiiiiiiiiiiiii
0.43 0.3 0.2 0.1 0.1-0.0-0.1-0.3-0.4iiiiiiiiiiiiiiiiiiiiiiiiiiii
0.44 0.3 0.2 0.2 0.1-0.0-0.1-0.2-0.4-0.6iiiiiiiiiiiiiiiiiiiiiiii
0.45 0.3 0.3 0.2 0.1 0.0-0.1-0.2-0.3-0.5iiiiiiiiiiiiiiiiiiiiiiii
0.46 0.4 0.3 0.2 0.2 0.1-0.0-0.2-0.3-0.5iiiiiiiiiiiiiiiiiiiiiiii
0.47 0.4 0.3 0.3 0.2 0.1 0.0-0.1-0.2-0.4-0.6iiiiiiiiiiiiiiiiiiii
0.48 0.4 0.4 0.3 0.2 0.1 0.0-0.1-0.2-0.4-0.6iiiiiiiiiiiiiiiiiiii
0.49 0.5 0.4 0.3 0.3 0.2 0.1-0.0-0.2-0.3-0.5iiiiiiiiiiiiiiiiiiii
0.50 0.5 0.4 0.4 0.3 0.2 0.1 0.0-0.1-0.3-0.4iiiiiiiiiiiiiiiiiiii
0.51 0.5 0.5 0.4 0.3 0.3 0.2 0.1-0.1-0.2-0.4-0.6iiiiiiiiiiiiiiii
0.52 0.6 0.5 0.5 0.4 0.3 0.2 0.1-0.0-0.2-0.3-0.5iiiiiiiiiiiiiiii
0.53 0.6 0.6 0.5 0.4 0.3 0.2 0.1 0.0-0.1-0.3-0.5iiiiiiiiiiiiiiii
0.54 0.6 0.6 0.5 0.5 0.4 0.3 0.2 0.1-0.1-0.2-0.4-0.7iiiiiiiiiiii
0.55 0.7 0.6 0.6 0.5 0.4 0.3 0.2 0.1-0.0-0.2-0.3-0.6iiiiiiiiiiii
0.56 0.7 0.7 0.6 0.5 0.5 0.4 0.3 0.2 0.0-0.1-0.3-0.5iiiiiiiiiiii
0.57 0.8 0.7 0.7 0.6 0.5 0.4 0.3 0.2 0.1-0.1-0.2-0.4-0.7iiiiiiii
0.58 0.8 0.8 0.7 0.6 0.6 0.5 0.4 0.3 0.1-0.0-0.2-0.4-0.6iiiiiiii
0.59 0.8 0.8 0.7 0.7 0.6 0.5 0.4 0.3 0.2 0.0-0.1-0.3-0.5iiiiiiii
0.60 0.9 0.8 0.8 0.7 0.6 0.6 0.5 0.4 0.2 0.1-0.1-0.2-0.4iiiiiiii
0.61 0.9 0.9 0.8 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.0-0.2-0.4-0.7iiii
0.62 1.0 0.9 0.9 0.8 0.7 0.7 0.6 0.5 0.3 0.2 0.1-0.1-0.3-0.6iiii
0.63 1.0 1.0 0.9 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.1-0.0-0.2-0.5iiii
0.64 1.1 1.0 1.0 0.9 0.8 0.8 0.7 0.6 0.4 0.3 0.2 0.0-0.2-0.4-0.7
0.65 1.1 1.1 1.0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.2 0.1-0.1-0.3-0.6
0.66 1.1 1.1 1.1 1.0 0.9 0.9 0.8 0.7 0.6 0.4 0.3 0.1-0.0-0.2-0.5

The i's represent imaginary territory.


A continuation of the program, arbitrarily choosing the column for c = 0.5 was progressively narrowed to find the difference x^2-y=0 (or y=x^2). The program in its narrowest state is:

c = .5
FOR b = .32738535# TO .32738537# STEP .0000000002#
  amb = 4 * b * b - (c - b) * (c - b)
  IF amb < 0 THEN
    PRINT "iiii"
  ELSE
    amb = SQR(amb)
    a = amb + b
    v = ((a + b) / 2) ^ 2 - (c + b) / 2
    PRINT USING "##.##########"; b; v; a; c
  END IF
NEXT

showing
                x^2 - y
b (=radius)   test value        a            c
0.3273853674-0.0000000001 0.9589935559 0.5000000000
0.3273853676 0.0000000003 0.9589935566 0.5000000000

where (a,b) and (b,c) are the coordinates of the centers of the circles.

That makes the centers at:

(0.9589935559,0.3273853674) and (0.3273853674,0.5)

the midpoint between these centers is at (.64318946165,.4136926837) where indeed y=x^2 and also the distances from each of the two centers are equal and equal the radius of each circle.

Similar solutions could be found for larger values of c. To take one, say c=1:

0.4696252803-0.0000000000 1.2447989847 1.0000000000
0.4696252803 0.0000000000 1.2447989848 1.0000000000

(1.2447989847,0.4696252803) and (0.4696252803,1) with radius 0.4696252803 satisfy the requirements.

And the pattern continues farther:

b\c  2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7
0.90 0.0-0.3iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.91 0.1-0.2-0.7iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.92 0.2-0.1-0.5iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.93 0.3 0.0-0.3iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.94 0.4 0.1-0.2-0.7iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.95 0.5 0.2-0.1-0.5iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.96 0.6 0.3 0.0-0.4iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.97 0.7 0.5 0.2-0.2-0.8iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.98 0.8 0.6 0.3-0.1-0.6iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
0.99 0.9 0.7 0.4 0.0-0.4iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
1.00 1.0 0.8 0.5 0.2-0.2iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
1.01 1.1 0.9 0.6 0.3-0.1-0.6iiiiiiiiiiiiiiiiiiiiiiiiiiii
1.02 1.2 1.0 0.7 0.4 0.0-0.4iiiiiiiiiiiiiiiiiiiiiiiiiiii
1.03 1.3 1.1 0.8 0.5 0.2-0.2iiiiiiiiiiiiiiiiiiiiiiiiiiii
1.04 1.4 1.2 0.9 0.6 0.3-0.1-0.7iiiiiiiiiiiiiiiiiiiiiiii
1.05 1.5 1.3 1.0 0.7 0.4 0.0-0.4iiiiiiiiiiiiiiiiiiiiiiii
1.06 1.6 1.4 1.1 0.8 0.5 0.2-0.3iiiiiiiiiiiiiiiiiiiiiiii
1.07 1.7 1.5 1.2 1.0 0.7 0.3-0.1-0.8iiiiiiiiiiiiiiiiiiii
1.08 1.8 1.6 1.3 1.1 0.8 0.4 0.0-0.5iiiiiiiiiiiiiiiiiiii
1.09 1.9 1.7 1.4 1.2 0.9 0.6 0.2-0.3iiiiiiiiiiiiiiiiiiii
1.10 2.0 1.8 1.5 1.3 1.0 0.7 0.3-0.1iiiiiiiiiiiiiiiiiiii
1.11 2.1 1.9 1.6 1.4 1.1 0.8 0.5 0.0-0.5iiiiiiiiiiiiiiii
1.12 2.2 2.0 1.8 1.5 1.2 0.9 0.6 0.2-0.3iiiiiiiiiiiiiiii
1.13 2.3 2.1 1.9 1.6 1.3 1.0 0.7 0.3-0.1iiiiiiiiiiiiiiii
1.14 2.4 2.2 2.0 1.7 1.5 1.2 0.8 0.5 0.0-0.6iiiiiiiiiiii
1.15 2.5 2.3 2.1 1.8 1.6 1.3 1.0 0.6 0.2-0.3iiiiiiiiiiii
1.16 2.6 2.4 2.2 1.9 1.7 1.4 1.1 0.8 0.4-0.1iiiiiiiiiiii
1.17 2.7 2.5 2.3 2.1 1.8 1.5 1.2 0.9 0.5 0.0-0.7iiiiiiii
1.18 2.8 2.6 2.4 2.2 1.9 1.6 1.3 1.0 0.6 0.2-0.4iiiiiiii
1.19 2.9 2.7 2.5 2.3 2.0 1.8 1.5 1.1 0.8 0.4-0.2iiiiiiii
1.20 3.0 2.8 2.6 2.4 2.1 1.9 1.6 1.3 0.9 0.5 0.0iiiiiiii
1.21 3.1 2.9 2.7 2.5 2.3 2.0 1.7 1.4 1.1 0.7 0.2-0.4iiii
1.22 3.2 3.0 2.8 2.6 2.4 2.1 1.8 1.5 1.2 0.8 0.4-0.2iiii
1.23 3.3 3.2 2.9 2.7 2.5 2.2 2.0 1.7 1.3 1.0 0.5 0.0iiii
1.24 3.5 3.3 3.1 2.8 2.6 2.3 2.1 1.8 1.5 1.1 0.7 0.2-0.5
1.25 3.6 3.4 3.2 3.0 2.7 2.5 2.2 1.9 1.6 1.2 0.8 0.4-0.2
1.26 3.7 3.5 3.3 3.1 2.8 2.6 2.3 2.0 1.7 1.4 1.0 0.6 0.0
1.27 3.8 3.6 3.4 3.2 3.0 2.7 2.4 2.2 1.8 1.5 1.1 0.7 0.2
1.28 3.9 3.7 3.5 3.3 3.1 2.8 2.6 2.3 2.0 1.6 1.3 0.9 0.4
1.29 4.0 3.8 3.6 3.4 3.2 2.9 2.7 2.4 2.1 1.8 1.4 1.0 0.6

with always, for a given c, a transition from imaginary to negative to positive, with the latter involving a location where x^2-y = 0.

Presumably what's wanted but not explicitly stated is that the circles should also be tangent to the parabola.


  Posted by Charlie on 2013-07-13 13:55:05
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