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Circle's Limit (Posted on 2006-01-13) Difficulty: 2 of 5
Let C1 be a circle with center A and radius 1. Let m be a line tangent to circle C1 at point B. Let C2 be a circle with center B intersecting line m at points C and D and circle C1 at points E and F (points labeled such that C and E are on the same side of line AB). Let line CE intersect line AB at point G. As the radius of circle C2 shrinks to zero, does the length of BG approach a limit? If yes, then what is its value?

See The Solution Submitted by Bractals    
Rating: 3.0000 (3 votes)

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I think I proved it without calculus. | Comment 1 of 4

Geometer's sketchpad implies the solution is 4.
Here's a possible analytic proof (but I feel like I broke the rules somewhere.  Or is this the calculus?)

Let c1 be (x-1)^2 + y^2 =1
Let c2 be x^2 + y^2 = r^2

C = (0,r)
E = (-r^2/2, r/2*sqrt(4-r^2))

[algebra omitted]
The line CE hits the x axis at x = -r^2/(2-sqrt(4-r^2))

2x - x*sqrt(4-r^2) = -r^2
r^2 + 2x = x*sqrt(4-r^2)
r^4 + 4xr^2 + 4x^2 = 4x^2 - x^2*r^2
x^2 + 4x + r^2 = 0 [it is legal to divide by r^2?]

If r=0 then

x^2 + 4x =0
Which has solutions

x=-4 or x=0

The 0 is extraneous, the x=-4 implies a distance of 4 from the origin.


  Posted by Jer on 2006-01-13 10:29:30
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