A parabola is the set of all points in the plane that are equidistant from a given line and a given point not on the line. A parabola is symmetric about the line that passes through the focus at right angles to the directrix.  This line, called the axis of the parabola, meets the parabola at a point called the vertex.

The given line is called the directrix of the parabola, and the given point the focus.

A parabola p can be defined as follows:

from five distinct points. The input is a list of five points. Note that a set of five distinct points does not necessarily define a parabola.

from the focus and vertex. The input is a list of the form ['focus'=fou, 'vertex'=ver] where fou and ver are explained above.

from the directrix and focus. The input is a list of the form ['directrix'=dir, 'focus'= fou] where dir and fou are explained above.

from its internal representation eqn. The input is an equation or a polynomial. If the optional argument n is not given, then:

if the two environment variables _EnvHorizontalName and _EnvVerticalName are assigned two names, these two names will be used as the names of the horizontal-axis and vertical-axis respectively.

if not, Maple will prompt for input of the names of the axes.

To access the information relating to a parabola p, use the following function calls:

The command with(geometry,parabola) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{geometry}\right)\:$

$\mathrm{parabola}\left(\mathrm{p1}\,y^2+12x-6y+33=0\,\left[x\,y\right]\right)\:$

$\mathrm{vertex}\left(\mathrm{p1}\right),\mathrm{coordinates}\left(\mathrm{vertex}\left(\mathrm{p1}\right)\right)$

$\mathrm{vertex\_p1},\left[\mathrm{-2}\,3\right]$

$\mathrm{focus}\left(\mathrm{p1}\right),\mathrm{coordinates}\left(\mathrm{focus}\left(\mathrm{p1}\right)\right)$

$\mathrm{focus\_p1},\left[\mathrm{-5}\,3\right]$

$\mathrm{directrix}\left(\mathrm{p1}\right),\mathrm{Equation}\left(\mathrm{directrix}\left(\mathrm{p1}\right)\right)$

$\mathrm{directrix\_p1},-1+x=0$

$\mathrm{parabola}\left(\mathrm{p2}\,\left['\mathrm{vertex}'=\mathrm{vertex}\left(\mathrm{p1}\right)\,'\mathrm{focus}'=\mathrm{focus}\left(\mathrm{p1}\right)\right]\,\left[x\,y\right]\right)\:$

$\mathrm{Equation}\left(\mathrm{p2}\right)$

$9y^2+108x-54y+297=0$

$\mathrm{parabola}\left(\mathrm{p3}\,\left['\mathrm{focus}'=\mathrm{focus}\left(\mathrm{p1}\right)\,'\mathrm{directrix}'=\mathrm{directrix}\left(\mathrm{p1}\right)\right]\,\left[x\,y\right]\right)\:$

$\mathrm{detail}\left(\mathrm{p3}\right)$

$\begin{array}{ll}\text{name of the object}& \mathrm{p3}\\ \text{form of the object}& \mathrm{parabola2d}\\ \text{vertex}& \left[\mathrm{-2}\,3\right]\\ \text{focus}& \left[\mathrm{-5}\,3\right]\\ \text{directrix}& -1+x=0\\ \text{equation of the parabola}& y^2+12x-6y+33=0\end{array}$

$\mathrm{point}\left(A\,-6\,3+4\mathrm{sqrt}\left(3\right)\right),\mathrm{point}\left(B\,-5\,9\right),\mathrm{point}\left(C\,-4\,3+2\mathrm{sqrt}\left(6\right)\right),\mathrm{point}\left(E\,-3\,3+2\mathrm{sqrt}\left(3\right)\right),\mathrm{point}\left(F\,-2\,3\right)\:$

$\mathrm{parabola}\left(\mathrm{p4}\,\left[A\,B\,C\,E\,F\right]\,\left[x\,y\right]\right)\:$

$\mathrm{eqn}≔\mathrm{Equation}\left(\mathrm{p4}\right)$

$\mathrm{eqn}≔\left(144\sqrt{3}\sqrt{2}-144\sqrt{2}-336\sqrt{3}+432\right)y^2+\left(1728\sqrt{3}\sqrt{2}-1728\sqrt{2}-4032\sqrt{3}+5184\right)x+\left(-864\sqrt{3}\sqrt{2}+864\sqrt{2}+2016\sqrt{3}-2592\right)y+4752\sqrt{3}\sqrt{2}-4752\sqrt{2}-11088\sqrt{3}+14256=0$

$\mathrm{lhs\_eqn}≔\mathrm{op}\left(1\,\mathrm{eqn}\right)\:$

$\mathrm{simplify}\left(\frac{\mathrm{lhs\_eqn}}{\mathrm{lcoeff}\left(\mathrm{lhs\_eqn}\,y\right)}\right)=\mathrm{op}\left(2\,\mathrm{eqn}\right)$

$y^2+12x-6y+33=0$

$\mathrm{radnormal}\left(\mathrm{coordinates}\left(\mathrm{vertex}\left(\mathrm{p4}\right)\right)\right)$

$\left[\mathrm{-2}\,3\right]$

$\mathrm{radnormal}\left(\mathrm{coordinates}\left(\mathrm{focus}\left(\mathrm{p4}\right)\right)\right)$

$\left[\mathrm{-5}\,3\right]$

$\mathrm{radnormal}\left(\mathrm{coordinates}\left(\mathrm{focus}\left(\mathrm{p4}\right)\right)\right)$

$\left[\mathrm{-5}\,3\right]$