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A unit circle has its center at (5,0) and a second circle with a radius of 2 units
has its center at (11,0) as shown. A common internal tangent to the circles intersects
the x-axis at Q(a,0). What is the value of a?
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Let A = (5,0) and E = (11,0) be the centers of the circles;
let B and D be the tangent points on greater and smaller circles, respectively;
Q = (a,0) is the intersection point of the common internal tangent to the circles
with the x-axis.
Draw the radii AB and ED to tangent points and consider triangles ABQ and QDE.
These triangles are right-angled triangles with right angles at the tangent points B and D.
They have congruent acute angles at the intersection point Q.
Hence, these triangles are similar.
Their corresponding sides are the radii AB = 1 and ED = 2 (the legs).
Their other corresponding sides are AQ and EQ (the hypotenuses),
so we can write
= = 2.
From the other side, AQ + EQ = AE = 11 - 5 = 6 units.
So, the intersection point Q divides the segment AE of the length 6 units
in segments EQ and AQ in proportion 2:1.
It implies that AQ is one third of AE=6; in other words, AQ = 2 units.
Hence, the value "a" under the problem's question is a = 5 + 2 = 7 units. ANSWER
Solved.