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 Post subject: Simple problem involving circlesPosted: Sun, 5 Aug 2007 22:03:42 UTC
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Question: Two different circles that passes through the point C(1,3) are tangent to both coordinate axes. If the length of a radius of the smaller circle is r and the length of a radius of the larger circle is R, what is the value of r+R?
Code:
Diagram:

o  o
y|    o        o
|
| o              o
|
|o        B       o     where B marks the center of the larger circle
|                             C marks the point of intersection of the two circles
| C* *           o            A marks the center of the smaller circle
|*  A  *
|*   o *      o
_| _*_*__o__o_________
O|
Spoiler:
The line that connects the centers of the two circles and is .
The slope of the line that connects the origin and is .
Let the angle (<COA = <COB) between and be .
Then
Therefore,
Code:
Diagram
|        . B
|     .   /
|   C   /
|   .\
|  .  A
| . /
_|./____ _
O|
Looking at triangle and triangle , , and .
For , by cosine law,
We have, or
Similarly, for ,
We have, or
The roots of the two equations for and are the same. That implies the smaller root equals and the larger root equals . Therefore the sum equals the sum of the roots which is given by

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 Post subject: Posted: Sun, 5 Aug 2007 23:36:08 UTC
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Joined: Fri, 20 Jul 2007 22:50:53 UTC
Posts: 50
Location: Texas
Spoiler:
Half of that proof seems unnecessary to me as only 2 circles tangent to both the x-axis and y-axis in the first quadrant will intersect a point (not on the axis) in quadrant I in the first place.

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 Post subject: AnswerPosted: Thu, 25 Sep 2008 08:42:52 UTC

Joined: Wed, 24 Sep 2008 12:44:04 UTC
Posts: 5
First observe that the coordinates of and are respectively and , therefore the equations of the circles are given by with for the small circle, etc. Now use the fact that belongs to both of them, then we get
for . Thus and it follows that .

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