First, remember that a tangent to a circle is perpendicular to a radial line segment to the point of tangency. Thus, recalling how we define the distance between a point and a line, the distance between a line tangent to a circle and the center of that circle is the radius of that circle. For a line of the form , the distance to the point is

Since the centers of the circles are (3,4) and (9,19), and the radii are 5 and 11, respectively, you have

and

Note you will have an unneeded degree of freedom (as you could multiply the line equation by a constant and still have the same line), so you can pick A,B,C to be scaled such that , so you then can define such that , and our equations are

and

Thus you have two equations for two unknowns. Note that there will be four solutions: two will be the 'external' tangents, and the other two will be 'internal' tangents (which cross each other between the circles)

--Kevin C.