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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?

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  • $\begingroup$ What do you mean by vertex? $\endgroup$ – Todor Markov Jan 5 at 17:33
  • $\begingroup$ Vertex of the parabola. $\endgroup$ – saket kumar Jan 5 at 17:36
  • $\begingroup$ If you know the axis, the tangent of the vertex is always perpendicular. Knowing it doesn't really give you anything new. $\endgroup$ – Todor Markov Jan 5 at 17:37
  • $\begingroup$ That's means only 1 parabola is possible as per condition $\endgroup$ – saket kumar Jan 5 at 17:39
  • $\begingroup$ No, you can make it as wide as you want, so infinitely many. You can also flip it upside down. $\endgroup$ – Todor Markov Jan 5 at 17:39
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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.

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