If 5 points are necessary to determine a conic, why are only 3 necessary to determine a parabola?

Question

I've just been reading about how 5 points are necessary and sufficient to determine a conic section in Euclidean geometry (https://en.wikipedia.org/wiki/Five_points_determine_a_conic). But if parabolas are conic sections, and if 3 points are sufficient to determine a parabola (since we can solve the resulting system of equations for the parabola equation's constant, coefficient of $x$, and coefficent of $x^2$), how can it be that 5 points are necessary to determine any conic section?

Answer 1

Thank you to Mikhail for suggesting that knowing from the beginning that the curve is a parabola allows you to determine the specific kind of parabola it is in less than five points. I believe this is the most satisfactory explanation, because given three points in a plane there is also a unique circle passing through them, since the circle equation is

$(x-a)^2+(y-b)^2 = r^2 $,

so knowing three points $(x, y)$ allows us to solve for $a, b$ and $r$.

Since circles are also conic sections, clearly three points are not enough to determine whether the curve is a parabola or a circle.

Some suggested more points are necessary to know the "direction/ orientation" of the parabola. There may be something to this. In fact, I believe the extended equation for parabolas (which may be rotated) is

$Ax^2+Bxy+Cy^2+Dx+Ey= F$,

so it seems that perhaps six points may be necessary to determine some parabolas? In any case, if we have that this is the form of the equation for a curve, we have not even eliminated the possibility that it is a circle until we show that either $B$, $D$, or $E$ $\neq{0}$, so this illustrates the first problem I mentioned: needing only three points was based on the initial assumption that we were dealing with a parabola.

All this raises the question of whether 5 points determine a conic section after you know you are dealing with a conic section in the first place, or before -- i.e., whether more points may be needed to determine that the curve defined by these 5 points is indeed a conic section and not some other thing altogether.

Answer 2

EDIT 1:

5 points for a conic. A parabola has $\epsilon=1, B^2 - 4 A C =0 $ that reduces number of constants/equations to 4.

$$ A x^2+ 2 \sqrt {A\,C} x y+C y^2+D x+E y= 1 $$

This can be cast into the form making $y$ subject of parabola equation :

$$ y = (a x + b) \pm \sqrt { c x + d }$$

If 3 constants are fixed, a single parameter family of parabolas is possible like in the example graphed :

$$ y = 2 x - 5 \pm \sqrt {3 x + 2 t} $$

Only 3 constants are not adequate to determine a general parabola .

Answer 3

As you can see from the image below, if you are defining a conic, it requires five points to be known. The conics below all share three points (black), but need two more to define what they are.

If we trying to simply define a parabola, it only takes three points because we know it is a parabola, and it can't possibly be a ellipse or hyperbola -- because we decided we already that we are making a parabola!

Answer 4

Illustrating a comment to OP's own answer.

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Shown is an entire family of parabolas through three points, indicating that three points alone do not determine a parabola.

As mentioned in my comment, a conic in the coordinate plane has five characteristics: eccentricity, scale, orientation, $x$-location, $y$-location. Our interest in parabolas specifies a characteristic (eccentricity $=1$); the points account for three more characteristics; and the animation cycles through the aspects of the fifth (here, orientation).