The equation of a line through the origin is of the form:Originally Posted by becky
such a line meets the parabola:
and if the line is a tangent this last equation has a single root
(since in general a line cuts a parabola at two points, one point
or no points, the one point case is a tangent).
Rearranging the equation gives:
and this has a single root when its discriminant is zero:
this is because a general quadratic equation has
roots , and both
of the roots are coincident (ie there is only one root) when
Solve this for and then plug these solutions back into the
equation of a line through the origin to get the equations for all
the the tangents to the parabola which pass through the origin.