Proof
(1) The line y = k cuts P where y = k and y2 = x, so x = k2.
Thus the line cuts the parabola only at (k2,k).
(2) Where the line meets the parabola, x = ny+c and y2 = x,
so that y2 = nx+c, i.e. y2-ny-c = 0. This quadratic has discriminant
n2+4c. Each root y = α of the quadratic leads to the point
(α2,α) on
the parabola. Hence
- the line cuts the parabola twice if n2+4c > 0,
- the line cuts the parabola at two coincident points if n2+4c = 0.
Thus the line is a tangent at this point.
- the line does not meet the conic if n2+4c < 0.
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