Hello, rednest!

First, find their intersection(s).Show that the line with equation

is a tangent to the ellipse with equation

Substitute [1] into [2]: .

. . which simplifies: .

. . and factors: .

. . and has one root: .

Substitute into [1] and we get: .

The line and ellipse have one common point: .

That should be sufficient, but let's check their slopes.

Obviously, the line has slope 2.

Differentiate [2] implicitly: .

At , the tangent to the ellipse has slope: .

. . Their slopes are equal!

Therefore, the line is tangent to the ellipse.