Relationship Between a Parabola and a Line

For the parabola and the line , the discriminant of the quadratic equation formed by eliminating determines their relative positions:

1. The line intersects the parabola at two distinct points.
2. The line is tangent to the parabola, touching at exactly one point.
3. The line does not intersect the parabola.

Equation of the Tangent Line to a Parabola

1. When the slope is given
   The equation of a line with slope tangent to the parabola is:
   The equation of a line with slope tangent to the parabola is:
   
   
2. When the coordinates of the point of tangency are given
   The equation of the tangent line to the parabola at the point is:
   The equation of the tangent line to the parabola at the point is: