2. Replace the term y in your equation of the ellipse by m(x-2)+3. You'll get a quadratic in x:
Solve for x.
3. Usually a straight line intercepts an ellipse in 2 or in 1 point or it is a passante that means there are no common points. The case that there exists only one point of interception occurs if the straight line is a tangent to the ellipse. This will happen if the discriminant equals zero:
Solve for m. You should come out with
4. Plug in these values into the equation of the line.