Show that the centers of the family of circles all of which are tangent to a given circle and also to a given line form a parabola.
Prove that the set of the centers of the family of circles all of which pass through a given point and are tangent to a given line forms a parabola.
Consider two parabolas which have the same focus and the same line as axis, and let these two parabolas have vertices lying on opposite sides of the focus. Prove that the parabolas intersect orthogonally (i.e., their tangent lines are perpendicular at the point of intersection).
The line is tangent to the parabola with equation . Find the point at which the line touches the parabola.