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I know that in order to uniquely determine a parabola, we require 3 points. So naturally, 2 points will have multiple possible parabolas pass through them.

My question is, how many?

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    $\begingroup$ Correction: In order to uniquely determine a parabola with an axis in a particular direction, we require 3 points. For why three points aren't sufficient in general, for instance, see, for instance, this answer. $\endgroup$ – Blue Jan 29 at 7:54
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    $\begingroup$ In questions like this, the possible answers are, zero, one and infinity. You already said it's more than one, so it must be infinity. $\endgroup$ – Oscar Bravo Jan 29 at 16:22
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Infinitely many. Draw a line between your two points. Now chose any points on the line as your third point. Since two different points on the line will give you different parabola, you get infinitely many different parabola.

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    $\begingroup$ Thank you! When you put it that way, it seems so obvious. Thanks again. $\endgroup$ – Dhruv Gupta Jan 29 at 7:57
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    $\begingroup$ You don't get a different parabola for every point. Still infinity, though. $\endgroup$ – Spitemaster Jan 29 at 12:53
  • $\begingroup$ You can set a third point anywhere not on any parabola you've yet considered. $\endgroup$ – Cai Jan 29 at 17:06
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    $\begingroup$ Yeah, naturally, you are right. I rewrote my answer. $\endgroup$ – Mundron Schmidt Jan 29 at 22:10
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Any size of parabola, with infinitely many orientations for each parabola!

Suppose the points are distance $d$ apart. Call them $A$ and $B$. Take your chosen parabola and pick two points on it separated by distance $d$. Then place them on $A$ and $B$.

Since you can slide the points along the parabola to wherever you like, the parabola can have any orientation that doesn't put its axis parallel to $AB$. (The sliding only gives you $180°$ worth of orientations, but you can reflect it in $AB$ to get the other $180°$.)

Assuming points on the parabola have real numbers as their coordinates, its size and orientation can be described by two real numbers: a size in the range $(0,\infty)$ and an angle in the range $(0,π)\cup(-π, 0)$.

Because of the way infinities work, this makes the set of parabolas through $A$ and $B$ uncountably infinite and the same size as the set of real nunbers.

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    $\begingroup$ +1 very pretty geometry. Much nicer than than the algebraic analysis. $\endgroup$ – Ethan Bolker Jan 29 at 14:24
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General equation of a conic section:

$\small {Ax^2+Bxy +Cy^2 +Dx+Ey +F=0}$;

where $A,B,C,D,E,F$ are constants.

Necessary condition for a parabola :

$B^2-4AC =0$, which leaves $5$ constants to be determined.

Cf. Blue's link in his comment above.

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