hyperbola - Maple Help

geometry

 hyperbola
 define a  hyperbola

 Calling Sequence hyperbola(p, [A, B, C, E, F], n) hyperbola(p, ['directrix'=dir, 'focus'=fou, 'eccentricity'=ecc], n) hyperbola(p, ['foci'=foi, 'vertices'=ver], n) hyperbola(p, ['foci'=foi, 'distancev'=disv], n) hyperbola(p, ['vertices'=ver, 'distancef'=disf], n) hyperbola(p, eqn, n)

Parameters

 p - the name of the hyperbola A, B, C, E, F - five distinct points 'directrix'=dir - dir is the line which is the directrix of the hyperbola 'focus'=fou - fou is a point which is the focus of the hyperbola 'eccentricity'=ecc - ecc is a constant bigger than one denoting the eccentricity of the hyperbola 'vertices'=ver - where ver is a list of two points which is the vertices of the hyperbola 'foci'=foi - foi is a list of two points which is the foci of the hyperbola 'distancev'=disv - where disv is the distance between the two vertices 'distancef'=disf - where disf is the distance between the two foci eqn - the algebraic representation of the hyperbola (i.e., a polynomial or an equation) n - (optional) a list of two names representing the names of the horizontal-axis and vertical-axis

Description

 • A hyperbola is the set of all points in the plane, the difference of whose distances from two fixed points is a given positive constant that is less than the distance between the fixed points.
 • The two fixed points are called the foci. The line through the foci is called the focal axis, and the line through the center and perpendicular to the focal axis is called the conjugate axis. The hyperbola intersects the focal axis at two points, called vertices.
 • Associated with every hyperbola is a pair of lines, called the asymptotes of the hyperbola. These lines intersect at the center of the hyperbola and have the property that as a point P moves along the hyperbola away from the center, the distance between P and one of the asymptotes approaches zero.
 • The two fixed points are called the foci.
 • A hyperbola 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 hyperbola.
 – from the directrix, focus, and eccentricity. The input is a list of the form ['directrix'=dir, 'focus'= fou, 'eccentricity' = ecc] where dir, fou, and ecc are explained above.
 – from the foci, and vertices. The input is a list of the form ['foci' = foi, 'vertices' = ver] where foi and ver are explained above.
 – from the foci and the distance between the two vertices. The input is a list of the form ['foci' = foi, 'distancev' = disv] where foi and disv are explained above.
 – from the vertices and the distance between the two foci. The input is a list of the form ['vertices' = ver, 'distancef' = disf] where ver and disf 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 an hyperbola p, use the following function calls:

 form(p) returns the form of the geometric object (i.e., hyperbola2d if p is a hyperbola). center(p) returns the name of the center of p. foci(p) returns a list of two foci of p. vertices(p) returns a list of two vertices of p. asymptotes(p) returns a list of two asymptotes of p. Equation(p) returns the equation that represents the hyperbola 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 hyperbola p.

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

Examples

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

define hyperbola h1 from its algebraic representation:

 > $\mathrm{hyperbola}\left(\mathrm{h1},9{y}^{2}-4{x}^{2}=36,\left[x,y\right]\right):$
 > $\mathrm{center}\left(\mathrm{h1}\right),\mathrm{coordinates}\left(\mathrm{center}\left(\mathrm{h1}\right)\right)$
 ${\mathrm{center_h1}}{,}\left[{0}{,}{0}\right]$ (1)
 > $\mathrm{foci}\left(\mathrm{h1}\right),\mathrm{map}\left(\mathrm{coordinates},\mathrm{foci}\left(\mathrm{h1}\right)\right)$
 $\left[{\mathrm{foci_1_h1}}{,}{\mathrm{foci_2_h1}}\right]{,}\left[\left[{0}{,}{-}\sqrt{{13}}\right]{,}\left[{0}{,}\sqrt{{13}}\right]\right]$ (2)
 > $\mathrm{vertices}\left(\mathrm{h1}\right),\mathrm{map}\left(\mathrm{coordinates},\mathrm{vertices}\left(\mathrm{h1}\right)\right)$
 $\left[{\mathrm{vertex_1_h1}}{,}{\mathrm{vertex_2_h1}}\right]{,}\left[\left[{0}{,}{-2}\right]{,}\left[{0}{,}{2}\right]\right]$ (3)
 > $\mathrm{asymptotes}\left(\mathrm{h1}\right),\mathrm{map}\left(\mathrm{Equation},\mathrm{asymptotes}\left(\mathrm{h1}\right)\right)$
 $\left[{\mathrm{asymptote_1_h1}}{,}{\mathrm{asymptote_2_h1}}\right]{,}\left[{y}{+}\frac{{2}{}{x}}{{3}}{=}{0}{,}{y}{-}\frac{{2}{}{x}}{{3}}{=}{0}\right]$ (4)

define hyperbola h2 from its foci and vertices:

 > $\mathrm{hyperbola}\left(\mathrm{h2},\left['\mathrm{vertices}'=\mathrm{vertices}\left(\mathrm{h1}\right),'\mathrm{foci}'=\mathrm{foci}\left(\mathrm{h1}\right)\right],\left[a,b\right]\right):$
 > $\mathrm{Equation}\left(\mathrm{h2}\right)$
 ${64}{}{{a}}^{{2}}{-}{144}{}{{b}}^{{2}}{+}{576}{=}{0}$ (5)

define hyperbola h3 from its foci and distance between the two vertices:

 > $\mathrm{hyperbola}\left(\mathrm{h3},\left['\mathrm{foci}'=\mathrm{foci}\left(\mathrm{h1}\right),'\mathrm{distancev}'=\mathrm{distance}\left(\mathrm{op}\left(\mathrm{vertices}\left(\mathrm{h1}\right)\right)\right)\right],\left[m,n\right]\right):$
 > $\mathrm{detail}\left(\mathrm{h3}\right)$
 $\begin{array}{ll}{\text{name of the object}}& {\mathrm{h3}}\\ {\text{form of the object}}& {\mathrm{hyperbola2d}}\\ {\text{center}}& \left[{0}{,}{0}\right]\\ {\text{foci}}& \left[\left[{0}{,}{-}\sqrt{{13}}\right]{,}\left[{0}{,}\sqrt{{13}}\right]\right]\\ {\text{vertices}}& \left[\left[{0}{,}{-2}\right]{,}\left[{0}{,}{2}\right]\right]\\ {\text{the asymptotes}}& \left[{n}{+}\frac{{2}{}{m}}{{3}}{=}{0}{,}{n}{-}\frac{{2}{}{m}}{{3}}{=}{0}\right]\\ {\text{equation of the hyperbola}}& {64}{}{{m}}^{{2}}{-}{144}{}{{n}}^{{2}}{+}{576}{=}{0}\end{array}$ (6)

define hyperbola h4 from its vertices and distance between the two foci:

 > $\mathrm{hyperbola}\left(\mathrm{h4},\left['\mathrm{vertices}'=\mathrm{vertices}\left(\mathrm{h1}\right),'\mathrm{distancef}'=\mathrm{distance}\left(\mathrm{op}\left(\mathrm{foci}\left(\mathrm{h1}\right)\right)\right)\right],\left[u,v\right]\right):$
 > $\mathrm{Equation}\left(\mathrm{h4}\right)$
 ${64}{}{{u}}^{{2}}{-}{144}{}{{v}}^{{2}}{+}{576}{=}{0}$ (7)

define hyperbola h5 from five distinct points:

 > $\mathrm{point}\left(A,1,\frac{2}{3}\mathrm{sqrt}\left(10\right)\right),\mathrm{point}\left(B,2,-\frac{2}{3}\mathrm{sqrt}\left(13\right)\right),\mathrm{point}\left(C,3,2\mathrm{sqrt}\left(2\right)\right),\mathrm{point}\left(E,4,-\frac{10}{3}\right),\mathrm{point}\left(F,5,\frac{2}{3}\mathrm{sqrt}\left(34\right)\right):$
 > $\mathrm{hyperbola}\left(\mathrm{h5},\left[A,B,C,E,F\right],\left[\mathrm{t1},\mathrm{t2}\right]\right):$

do some simplifications:

 > $\mathrm{remove}\left(\mathrm{type},\mathrm{radnormal}\left(\mathrm{op}\left(1,\mathrm{Equation}\left(\mathrm{h5}\right)\right)\right),\mathrm{constant}\right)$
 ${4}{}{{\mathrm{t1}}}^{{2}}{-}{9}{}{{\mathrm{t2}}}^{{2}}{+}{36}$ (8)

define hyperbola h6 from its directrix, focus and eccentricity:

 > $\mathrm{line}\left(l,x=-2,\left[x,y\right]\right):$$\mathrm{point}\left(f,1,0\right):$$e≔\frac{3}{2}:$
 > $\mathrm{hyperbola}\left(\mathrm{h6},\left['\mathrm{directrix}'=l,'\mathrm{focus}'=f,'\mathrm{eccentricity}'=e\right],\left[c,d\right]\right):$
 > $\mathrm{eq}≔\mathrm{Equation}\left(\mathrm{h6}\right)$
 ${\mathrm{eq}}{≔}{-}\frac{{5}}{{4}}{}{{c}}^{{2}}{-}{11}{}{c}{+}{{d}}^{{2}}{-}{8}{=}{0}$ (9)