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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)?

 Dec 20, 2018
 #1
avatar+98044 
+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!!!!

 

 

 

cool coolcool

 Dec 20, 2018
 #2
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0

Sorry, that isn't correct

Guest Dec 20, 2018
 #3
avatar+98044 
0

OK.....maybe someone else can shed some light on this....!!!

 

 

cool cool cool

CPhill  Dec 20, 2018

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