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
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