Y by
Two straight pipes (circular cylinders), with radii 
and 
, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?
![[asy]
size(6cm);
draw(circle((0,1),1), linewidth(1.2));
draw((-1,0)--(1.25,0), linewidth(1.2));
draw(circle((1,1/4),1/4), linewidth(1.2));
[/asy]](//latex.artofproblemsolving.com/4/0/9/4091eff69445abb83b260b3ccbdfa4e00b8f9182.png)
and 
, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?![[asy]
size(6cm);
draw(circle((0,1),1), linewidth(1.2));
draw((-1,0)--(1.25,0), linewidth(1.2));
draw(circle((1,1/4),1/4), linewidth(1.2));
[/asy]](http://latex.artofproblemsolving.com/4/0/9/4091eff69445abb83b260b3ccbdfa4e00b8f9182.png)
This post has been edited 5 times. Last edited by mathboy282, Nov 13, 2024, 6:40 PM
by Descartes Theorem.
is the desired radius let 
,
,
.
,
.
will be the smaller radius, given that it is an answer choice, along with the fact that (C) and (D) differ by 
![[asy]
size(6cm);
draw(circle((0,1),1), linewidth(1.2));
draw((-1,0)--(1.25,0), linewidth(1.2));
draw(circle((1,1/4),1/4), linewidth(1.2));
[/asy]](http://latex.artofproblemsolving.com/d/f/b/dfb2481a375b7f929011e76a6fc8c8fb09be835f.png)
![[asy]
size(6cm);
draw(circle((0,1),1), linewidth(1.2));
draw(circle((2,1),1), linewidth(1.2));
draw(circle((0.666 , 0.111),0.111), linewidth(1.2));
draw((-1,0)--(3.25,0), linewidth(1.2));
draw(circle((1,1/4),1/4), linewidth(1.2));
[/asy]](http://latex.artofproblemsolving.com/6/1/d/61d0f8545672d748fb9dd0113ed3684f20787fb2.png)
.
so 
or 
.
where 
is the smallest circle of the 
circles of radii 
and a third circle of radius 
that is externally tangent to the first two circles as well as one of their common external tangents, then we have 
Applying this formula with 
as well as 
we get 
and 
so the answer is 
and let the center of the second circle be 
.![\[
x^2 + (y-1)^2 = (y + 1)^2
\]](http://latex.artofproblemsolving.com/9/e/e/9ee71a190ba030ed6c6509b5eacdf50bbf0f27e4.png)
and the second circle consists of![\[
(x-1)^2 + (y-\frac14)^2 = (y + \frac{1}{4})^2
\]](http://latex.artofproblemsolving.com/4/c/6/4c61ce37a9b9b100ac8dd668ddcfc810365d0575.png)
Then we can solve this as a quadratic in 
to get 
and thus 
are the radii in a configuration like this, then 
Proof: 2015 AMC 12A #25.
giving 
