proceed in two phases
Step 1: Find the slope of the tangent.
From the knowledge of plane geometry, we can find the included angle between the outer common tangent and the connecting line.
The radius of the tangent point perpendicular to the two tangents through an outer tangent.
And the center distance.
Form a right-angled trapezoid
In this trapezoid, the angle between the outer common tangent and the connecting line can be found.
Then from the included angle formula
Slope through the connecting line
Find the slopes of two tangents.
Step 2: Find the coordinates of a point on a straight line.
According to insiders' understanding,
You can know
A point where two tangents intersect on the center line. Let's find the coordinates of this point.
Two triangles consisting of a tangent line, a connecting line and two radii, the ratio of the distance between this point and the two center points can be obtained from the knowledge of proportional line segments; Then, the coordinates of the fixed point are calculated by the coordinate formula of the fixed point.
Finally, the straight line equation is obtained from the oblique shape of the straight line point.
There are too many calculation processes to write here. Explain the train of thought, you should be able to do it.