Two circles of radius 1 are centered at (4,0) and (-4,0) How many circles are tangent to both of the given circles and also pass through the point (0,5)?

AoPS.Morrisville Nov 11, 2019

#1**+1 **

*Two circles of radius 1 are centered at (4,0) and (-4,0) How many circles are tangent to both of the given circles and also pass through the point (0,5)?*

I'm going to say 4. A circle can be drawn through three points, so the circles will pass through (0,5) and the following:

1. Tangent to both the small circles at points farther from the origin.

2. Tangent to both the small circles at points nearer to the origin.

3. Tangent to the left circle at a point nearer to the origin and the right circle at a point farther from the origin.

4. Tangent to the left circle at a point farther from the origin and the right circle at a point nearer to the origin.

The problem did not require that the solution circle(s) be symmetrical around either axis.

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Guest Nov 11, 2019