Why doesn't discriminant notice a solution that has the same slope as the axis of symmetry of a parabola?

Question

Hard to word out the problem, so here's an example.

Say I wanted to find out the lines from a point (-1, 0) which have only one intersection with a parabola $x=y^2$ All the lines from (-1, 0) are $y=kx+k$ where k is the slope.

Combining $x=y^2$ and $y=kx+k$ to find the formula for the intersections, ultimately gives us $0=k^2×x^2+(2k-1)x+k^2$ Now solving the discriminant to be zero we get that k has to be either 1/2 or -1/2. Placing the values to the line formula, we of course get the tangents of the parabola. But there's also a third answer: k=0. Then we get a line $y=0$ which also has only one intersection with $x=y^2$ at (0,0).

Why doesn't discriminant notice this solution?

Answer 1

Because when $k = 0$, the equation is linear, not quadratic.