parabola - Maple Help

geometry

 parabola
 define a parabola

 Calling Sequence parabola(p, [A, B, C, E, F], n ) parabola(p, ['focus'=fou, 'vertex'=ver], n ) parabola(p, ['directrix'=dir, 'focus'=fou], n ) parabola(p, eqn, n )

Parameters

 p - the name of the parabola A, B, C, E, F - five distinct points 'focus'=fou - fou is the point which is the focus of the parabola 'vertex'=ver - ver is the point which is the vertex of the parabola 'directrix'=dir - dir is the line which is the directrix of the parabola eqn - the algebraic representation of the parabola (i.e., a polynomial or an equation) n - (optional) list of two names representing the names of the horizontal-axis and vertical-axis

Description

 • 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:

 form(p) returns the form of the geometric object (i.e., parabola2d if p is a parabola). vertex(p) returns the name of the vertex of p. focus(p) returns the name of the focus of p. directrix(p) returns the name of the directrix of p. Equation(p) returns the equation that represents the parabola p. HorizontalName(p) returns the name of the horizontal-axis; or FAIL if the axis is not assigned a name. VerticalName(p) returns the name of the vertical-axis; or FAIL if the axis is not assigned a name. detail(p) returns a detailed description of the parabola p.

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

Examples

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

define parabola p1 from its algebraic representation:

 > $\mathrm{parabola}\left(\mathrm{p1},{y}^{2}+12x-6y+33=0,\left[x,y\right]\right):$
 > $\mathrm{vertex}\left(\mathrm{p1}\right),\mathrm{coordinates}\left(\mathrm{vertex}\left(\mathrm{p1}\right)\right)$
 ${\mathrm{vertex_p1}}{,}\left[{-2}{,}{3}\right]$ (1)
 > $\mathrm{focus}\left(\mathrm{p1}\right),\mathrm{coordinates}\left(\mathrm{focus}\left(\mathrm{p1}\right)\right)$
 ${\mathrm{focus_p1}}{,}\left[{-5}{,}{3}\right]$ (2)
 > $\mathrm{directrix}\left(\mathrm{p1}\right),\mathrm{Equation}\left(\mathrm{directrix}\left(\mathrm{p1}\right)\right)$
 ${\mathrm{directrix_p1}}{,}{-}{1}{+}{x}{=}{0}$ (3)

define parabola p2 from its focus and vertex

 > $\mathrm{parabola}\left(\mathrm{p2},\left['\mathrm{vertex}'=\mathrm{vertex}\left(\mathrm{p1}\right),'\mathrm{focus}'=\mathrm{focus}\left(\mathrm{p1}\right)\right],\left[x,y\right]\right):$
 > $\mathrm{Equation}\left(\mathrm{p2}\right)$
 ${9}{}{{y}}^{{2}}{+}{108}{}{x}{-}{54}{}{y}{+}{297}{=}{0}$ (4)

define parabola p3 from its directrix and its focus

 > $\mathrm{parabola}\left(\mathrm{p3},\left['\mathrm{focus}'=\mathrm{focus}\left(\mathrm{p1}\right),'\mathrm{directrix}'=\mathrm{directrix}\left(\mathrm{p1}\right)\right],\left[x,y\right]\right):$
 > $\mathrm{detail}\left(\mathrm{p3}\right)$
 $\begin{array}{ll}{\text{name of the object}}& {\mathrm{p3}}\\ {\text{form of the object}}& {\mathrm{parabola2d}}\\ {\text{vertex}}& \left[{-2}{,}{3}\right]\\ {\text{focus}}& \left[{-5}{,}{3}\right]\\ {\text{directrix}}& {-}{1}{+}{x}{=}{0}\\ {\text{equation of the parabola}}& {{y}}^{{2}}{+}{12}{}{x}{-}{6}{}{y}{+}{33}{=}{0}\end{array}$ (5)

define parabola p4 from five distinct points

 > $\mathrm{point}\left(A,-6,3+4\mathrm{sqrt}\left(3\right)\right),\mathrm{point}\left(B,-5,9\right),\mathrm{point}\left(C,-4,3+2\mathrm{sqrt}\left(6\right)\right),\mathrm{point}\left(E,-3,3+2\mathrm{sqrt}\left(3\right)\right),\mathrm{point}\left(F,-2,3\right):$
 > $\mathrm{parabola}\left(\mathrm{p4},\left[A,B,C,E,F\right],\left[x,y\right]\right):$
 > $\mathrm{eqn}≔\mathrm{Equation}\left(\mathrm{p4}\right)$
 ${\mathrm{eqn}}{≔}\left({144}{}\sqrt{{3}}{}\sqrt{{2}}{-}{144}{}\sqrt{{2}}{-}{336}{}\sqrt{{3}}{+}{432}\right){}{{y}}^{{2}}{+}\left({1728}{}\sqrt{{3}}{}\sqrt{{2}}{-}{1728}{}\sqrt{{2}}{-}{4032}{}\sqrt{{3}}{+}{5184}\right){}{x}{+}\left({-}{864}{}\sqrt{{3}}{}\sqrt{{2}}{+}{864}{}\sqrt{{2}}{+}{2016}{}\sqrt{{3}}{-}{2592}\right){}{y}{+}{4752}{}\sqrt{{3}}{}\sqrt{{2}}{-}{4752}{}\sqrt{{2}}{-}{11088}{}\sqrt{{3}}{+}{14256}{=}{0}$ (6)
 > $\mathrm{lhs_eqn}≔\mathrm{op}\left(1,\mathrm{eqn}\right):$
 > $\mathrm{simplify}\left(\frac{\mathrm{lhs_eqn}}{\mathrm{lcoeff}\left(\mathrm{lhs_eqn},y\right)}\right)=\mathrm{op}\left(2,\mathrm{eqn}\right)$
 ${{y}}^{{2}}{+}{12}{}{x}{-}{6}{}{y}{+}{33}{=}{0}$ (7)
 > $\mathrm{radnormal}\left(\mathrm{coordinates}\left(\mathrm{vertex}\left(\mathrm{p4}\right)\right)\right)$
 $\left[{-2}{,}{3}\right]$ (8)
 > $\mathrm{radnormal}\left(\mathrm{coordinates}\left(\mathrm{focus}\left(\mathrm{p4}\right)\right)\right)$
 $\left[{-5}{,}{3}\right]$ (9)
 > $\mathrm{radnormal}\left(\mathrm{coordinates}\left(\mathrm{focus}\left(\mathrm{p4}\right)\right)\right)$
 $\left[{-5}{,}{3}\right]$ (10)