# Parabola through 3 points

(from Didactic hints)

In the Cartesian plane xOy, equations like

with a≠0, represent parabolas with axis of symmetry parallel to the y-axis and equations like

represent parabolas with axis of symmetry parallel to the x-axis.

As in (1) and (2) there are three parameters, three independent conditions are sufficient to determine the equations of the parabolas of either type.

In particular, given three points A, B, C such that the segments AB and BC are not parallel to the axes:

the equation of the parabola that contains them can be obtained by requiring that the coordinates of the points satisfy the equation of the parabola. For parabolas of type (1) we have

and for parabolas of type (2)

The following Javascript application automates the solution of the problem. When the coordinates of the points are rational (ie do not contain the decimal point but can contain a slash) also the parameters a, b and c are expressed by rational numbers.

The application calculates equation and vertex and produces the graph of the parabola in the area next to vertex. The maximum distance from the focus must be written in the input field ` max. ρ .`

``` last revision: September 2018 ```