1. Calculate the coordinates of the points of intersection. If the line is actually a tangent to the ellipse you only get one point of intersection.

2.

Plug in the term for y into the 2nd equation and solve for x.

3.

Expand the bracket and collect like terms:

This is a quadratic equation in x. Use the quadratic formula and solve for x:

4. You only get one common point of the line and the ellipse (= tangent point) if the radical equals zero: