There are two circles, one with center A and radius 1, the other with center B and radius 2. The distance AB is 6. A third circle of unknown radius is tangent to both of these circles and there exists a straight line which

(1) is tangent to all three circles, and

(2) intersects the segment AB.

Find the radius of the third circle.

There are two solutions, R and r. Evaluate R - r.

Guest Jul 5, 2020

#1**+1 **

There are two solutions, R and r. Evaluate R - r.

**Hello Guest!**

The distance \(\overline{AB}\) intersected by the tangent is divided into a and b.

\(a:(6-a)=r_A:r_B\\ a:(6-a)=1:2\\ 2a=6-a\\ a=2\\ b=4\)

\(r_A:(r_A+R)=a:\overline{AB}\\ 1:(1+R)=2:6\\ 6=2+2R \)

\(R=2\)

The radius of the third circle is 2.

There are not two solutions R.

!

asinus Jul 5, 2020