klein view page

tangents to a parabola

The Parabola P : y² = x.
(1) The line y = k cuts P exactly once.
(2) The line x = ny+c

cuts P twice if n²+4c > 0,
is a tangent to P if n²+4c = 0, and
does not meet P if n²+4c < 0.

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.

special conics page

The Parabola P : y² = x. (1) The line y = k cuts P exactly once. (2) The line x = ny+c cuts P twice if n²+4c > 0, is a tangent to P if n²+4c = 0, and does not meet P if n²+4c < 0.
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.