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 Circles & Parabola (Posted on 2013-07-13)

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 No Rating 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 Solution Harry 2013-07-14 06:43:52 computer approximation Charlie 2013-07-13 17:07:42 a start Charlie 2013-07-13 15:46:14 re: thoughts - Problem Correction Bractals 2013-07-13 14:30:53 thoughts Charlie 2013-07-13 13:55:05
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