Maximum number of parabolas that can be drawn with a given axis and tangent at vertex.

Question

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?

If you know the axis, the tangent of the vertex is always perpendicular. Knowing it doesn't really give you anything new. — Todor Markov, Jan 05 '19 at 17:37
No, you can make it as wide as you want, so infinitely many. You can also flip it upside down. — Todor Markov, Jan 05 '19 at 17:39
If we flip the parabola then the equation of the axis gets changed . I think so. — Tips, Jan 05 '19 at 17:42
What do you mean by equation of the axis? If it's the standard $y = ax + b$, then it's the equation of the line, which doesn't have a direction. So you can flip. — Todor Markov, Jan 05 '19 at 17:46
@saket kumar If you flip the parabola without touching the axis, why should the equation of that axis get changed? There are two positions or possibilities no matter whether the axis is parallel to one of the coordinate axes ot not. — Narasimham, Jan 06 '19 at 00:02

score 2 · Accepted Answer · answered Jan 05 '19 at 17:48

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.

Maximum number of parabolas that can be drawn with a given axis and tangent at vertex.

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