Proof
(1) The line y = k cuts P where y = k and y^{2} = x, so x = k^{2}.
Thus the line cuts the parabola only at (k^{2},k).
(2) Where the line meets the parabola, x = ny+c and y^{2} = x,
so that y^{2} = nx+c, i.e. y^{2}nyc = 0. This quadratic has discriminant
n^{2}+4c. Each root y = α of the quadratic leads to the point
(α^{2},α) on
the parabola. Hence
 the line cuts the parabola twice if n^{2}+4c > 0,
 the line cuts the parabola at two coincident points if n^{2}+4c = 0.
Thus the line is a tangent at this point.
 the line does not meet the conic if n^{2}+4c < 0.

