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)?

Guest Dec 20, 2018

#1**+1 **

Here's my best attempt.....

I believe that the answer is 1

When two circles are tangent....a line drawn through their centers will pass through the point of tangency...so....let the center of the the circle be (x,y) and the radius = r

The distance for the center of the circle to (0,5) will just = r

The distance from the center of this circle to the centers of the given circles will be r + 1

We have this system

(x - 4)^2 + y^2 = (r + 1)^2

(x + 4)^2 + y^2 = (r + 1)^2

x^2 + (y - 5)^2 = r^2

This is a little messy to solve [ but not impossible]...so I used WolframAlpha to generate the solution

The center of the circle is (x, y) = (0, 5/3) and r = 10/3

Here's a graph :

PS....I can show you how to solve the equations if need be...but it's a little lengthy!!!!

CPhill Dec 20, 2018