conic sections
If the equation of axis and the tangent at vertex are given, then what is the maximum number of parabolas that can be drawn?
My approach is this: Since the equation of axis and tangent at the vertex is fixed, then only 1 parabola is possible. Am I right? Or are there infinitely-many parabolas that can drawn by the given condition?
To find the vertex of that parabola, you can make use of a nice general result:
given three tangents of a parabola, if $AB$ is that part of a tangent which is comprised between the other two tangents (see diagram), then the projection of $AB$ onto a line perpendicular to the axis has a fixed length, independent of the position of $AB$.
In our particular case, if $V$ (vertex) and $D$ are the tangency points of the "outer" tangents, $C$ is their intersection point, and $B'$, $D'$ are the projections of $B$, $D$ on tangent $VC$ (which is perpendicular to the axis), we obtain from the above result:
$$AB'=CD'=VC.$$
As points $A$, $B$, $C$, $B'$ can be readily found from the data, one can also easily find vertex $V$ and tangency point $D$.
If the directrix and the tangent line at the vertex are given but not the vertex, there exist infinitely many parabolas.
The vertex of such parabola is any point lying at this tangent line.
The width of all these parabolas is the same, because is uniquely determined by the distance between the two given lines ( they are parallel).
Best Answer
The tangent at the vertex is always perpendicular to the axis. So, if you know the axis and the tangent at the vertex, it's essentially the same as knowing the axis and the vertex only. So you can make your parabola arbitrarily wide, and you can also flip it, so essentially you have infinitely many parabolas.
If, instead, you have the axis, a point not on the axis (i.e. not the vertex), and a tangent to that point, then in general you'd have a unique parabola.