Given a fixed parabola and a family of parallel lines with given fixed gradient, find the one line that intersects the parabola in one single point

Let the parabola be y=x² and let the family of lines be y=2x+c, in order to study the intersection points we need to consider the second order linear system given by having the two equations above. Hence, we get x² -2x-c=0 and this equation has one single solution if and only if -c=1.

Therefore, the solution line is y=2x-1