Mensa Tangent Line Algebra & Geometry Solution.

 

                                         Elaborated tangent line and circles diagram

The problem is easily solved by first doing a smaller triangle construction such as shown in the diagram.  Here we designate L as the line that passes through the centers of the two circles.  We then designate CTL as the common tangent line, i.e. on the side opposite the x-axis.

We then designate  q_L as the angle between line L and the x-axis.  In like manner we designate q_CTL as the angle between line CTL and the x-axis.

From inspection,  tan (q_L ) = 9/40, i.e. the slope between the centers of the two circles.   But note that line L bisects the angle created by the4 x-axis and the line CTL.  Hence we can write:  

q_CTL  = 2 q_L

By the tangent double angle formula:

tan (2 q_L) =  2 tan q_L /   (1 - tan ² q_L ) =

2 (9/40) / (1 - 81/ 1600)  = 18/40 / (1519/1600)

=  720/ 1519   

Which is the slope m of the line CTL

Lines L and CTL intersect the x-axis at the same point.

The slope formula is y = mx + b 

Solve for b using the line L, i.e.

16 =  (9/40) 4 + b

à

b = 16  -  (36/40) = 151/ 10

Then the equation of the common tangent line (on the opposite side of the x-axis) is:

y = ( 720 x/ 1519) + 151/10