Question 1066762
I hope you live in California, because it is past 11 PM on the East coast.
I am guessing what the meaning of yoir question is.
I think you are talking about a line tangent to two circles with centers 12 units apart,
with radius 5 and 3 units.
The circles, with a tangent line, the radii to the tangency points,
and the line going through both centers are drawn below.
I assume A and B are the tangency points,
and you want to find the distance between them.
{{{drawing(360,140,-5.5,30.5,-2,12,
line(-6,0,36,0),
circle(0,5,5),circle(12,3,3),
red(triangle(0,0,0,5,30,0)),red(triangle(30,0,12,0,12,3)),
red(rectangle(0,0,0.7,0.7)),red(rectangle(12,0,12.7,0.7)),
locate(0.3,3,red(5)),locate(12.3,2.3,red(3)),
locate(6,4,red(12)),locate(18,2.2,red(x)),
locate(-0.2,0,A),locate(11.8,0,B),locate(29.8,0,P)
)}}}
In the drawing you see two similar right triangles.
The short legs measure 5, and 3 (the circles' radii).
Because the triangles are similar,
the ratio of hypotenuse to long leg is similar for both, so
{{{x/3=(12+x)/5}}}--->{{{5x=3(12+x)}}}--->{{{5x=36+3x}}}--->{{{5x-3x=36}}}--->{{{2x=36}}} ---> {{{x=36/2}}}--->{{{x=18}}}
Know that we know {{{x}}}, we can use the Pythagorean theorem,
with the hypotenuse and short leg of both triangles to found the long legs.
For the small triangle:
{{{BP^2+3^2=18^2}}}-->{{{BP^2+9=324}}}-->{{{BP^2=324-9}}}-->{{{BP^2=315}}}-->{{{BP=sqrt(315)=SQRT(9*35)=3SQRT(35)}}} .
For the large triangle:
{{{AP^2+5^2=(12+18)^2}}}-->{{{AP^2+5^2=30^2}}}-->{{{AP^2+25=900}}}-->{{{AP^2=900-25}}}-->{{{AP^2=875}}}-->{{{AP=sqrt(875)=sqrt(25*35)=5sqrt(35)}}} .
So,
{{{AB=AP-BP=5sqrt(35)-3sqrt(35)=highlight(2sqrt(35))}}} .
Did I guess right?