Parabola max. number

Question

If the directrix and the tangent at vertex of a parabola are given then what is the maximum number of parabolas that can be drawn? Well according to me the answer should be 1 because the distance between tangent at vertex and directrix is a constant (=a).But the solution says infinity .I searched on Google plus my textbooks but they are of no help.

Answer 1

Any parabola with a given directrix that is tangent to a given line at its vertex can be translated in a direction parallel to the two lines giving another such parabola. Thus there are infinitely many parabolas, though they are all congruent.

EDIT: The parabolas are all congruent because of the definition of a parabola as the locus of points equidistant from a certain point and line, respectively, focus point and directrix. Let the distance between the given directrix and tangent line be $a$. By the definition of a parabola, we know that the focus point is at a distance of $2a$ from the directrix. Thus, once we pick a point on the tangent line to be our vertex, our parabola is uniquely defined.