First the line is tangent to the ellipse and passes through the point (4,6).
If there is another one its equation is
It is tangent to the ellipse. Therefore there is only one contact point between the ellipse and the line. In other words the system
has one single solution
It means that the equation or has one solution. Its discriminant is thus equal to 0.
The line passes through the point (4,6). Therefore or .
Then giving and