TITLE: If 5 points are necessary to determine a conic, why are only 3 necessary to determine a parabola?
QUESTION [15 upvotes]: 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?

REPLY [27 votes]: Illustrating a comment to OP's own answer.


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).