Formulas and Tables

Analytic Geometry

Hyperbola and Parabola

Transverse axis of a hyperbola: \(2a\)
Conjugate axis of a hyperbola: \(2b\)
Real numbers: \(A,\) \(B,\) \(C,\) \(D,\) \(E,\) \(F,\) \(t\)
Point coordinates: \(x\), \(y\)
Focal distance: \(2c\)

Foci of a hyperbola: \({F_1}\), \({F_2}\)
Distance from any point of a hyperbola to its foci: \({r_1},\) \({r_2}\)
Eccentricity of a hyperbola: \(e\)
Focal parameter of a parabola: \(p\)
Focus of a parabola: \(F\)

  1. A hyperbola is a plane curve such that the difference of the distances from any point of the curve to two other fixed points (called the foci of the hyperbola) is constant. The distance between the foci of a hyperbola is called the focal distance and denoted as \(2c\). Any hyperbola consists of two distinct branches. The points on the two branches that are closest to each other are called the vertices. The line segment joining the vertices of a hyperbola is called the transverse axis and usually denoted as \(2a\). The midpoint of this line segment is the hyperbola’s center. The transverse axis of a hyperbola is its line of symmetry. Another line of symmetry is perpendicular to the transverse axis and is called the conjugate axis. Its length is denoted as \(2b\). The canonical equation of a hyperbola in the Cartesian coordinate system is written in the form
    \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} – {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} = 1\).
A hyperbola
  1. The absolute value of the difference of the distances from any point of a hyperbola to its foci is constant:
    \(\left| {{r_1} – {r_2}} \right| = 2a\),
    where \({r_1}\), \({r_2}\) are the distances from an arbitrary point \(P\left( {x,y} \right)\) of the hyperbola to the foci \({F_1}\) and \({F_2}\), \(a\) is the transverse semi-axis of the hyperbola.
The absolute value of the difference of the distances from any point of a hyperbola to its foci is constant
  1. Equations of the asymptotes of a hyperbola
    \(y = \pm {\large\frac{b}{a}\normalsize} x\)
  2. Relationship between semi-axes of a hyperbola and its focal distance
    \({c^2} = {a^2} + {b^2}\),
    where \(c\) is half the focal distance, \(a\) is the transverse semi-axis of the hyperbola, \(b\) is the conjugate semi-axis.
  3. Eccentricity of a hyperbola
    \(e = {\large\frac{c}{a}\normalsize} \gt 1\)
  4. Equations of the directrices of a hyperbola
    The directrix of a hyperbola is a straight line perpendicular to the transverse axis of the hyperbola and intersecting it at the distance \(\large\frac{a}{e}\normalsize\) from the center. A hyperbola has two directrices spaced on opposite sides of the center. The equations of the directrices are given by
    \(x = \pm {\large\frac{a}{e}\normalsize} = \pm {\large\frac{{{a^2}}}{c}\normalsize}\).
  5. Parametric equations of the right branch of a hyperbola
    \(
    \left\{
    \begin{aligned}
    x &= a \cosh t \\
    y &= b \sinh t
    \end{aligned}
    \right.,\;\) \(0 \le t \le 2\pi\),
    where \(a\), \(b\) are the semi-axes of the hyperbola, \(t\) is a parameter.
  6. General equation of a hyperbola
    \(A{x^2} + Bxy \) \(+\; C{y^2} + Dx\) \(+\; Ey + F = 0,\)
    where \(B^2 – 4AC \gt 0\).
  7. General equation of a hypebola with axes parallel to the coordinate axes
    \(A{x^2} + C{y^2} \) \(+\; Dx + Ey \) \(+\; F = 0,\)
    where \(AC \lt 0\).
  8. Equilateral hyperbola
    A hyperbola is called equilateral it its semi-axes are equal to each other: \(a = b\). Such a hyperbola has mutually perpendicular asymptotes. If the asymptotes are taken to be the horizontal and vertical coordinate axes (respectively, \(y = 0\) and \(x = 0\)), then the equation of the equilateral hyperbola has the form
    \(xy = {\large\frac{{{e^2}}}{4}\normalsize}\)  or  \(y = {\large\frac{k}{x}\normalsize}\), where \(k = {\large\frac{e^2}{4}\normalsize} .\)
Equilateral hyperbola
  1. A parabola is a plane curve, every point of which has the property that the distance to a fixed point (called the focus of the parabola) is equal to the distance to a straight line (the directrix of the parabola). The distance between the focus to the directrix is called the focal parameter and denoted by \(p.\) A parabola has a single axis of symmetry that intersects the parabola at its vertex. The canonical equation of a parabola has the form
    \(y = 2px\).

    Equation of the directrix
    \(x = – {\large\frac{p}{2}\normalsize}\),
    where \(p\) is the focal parameter.

    Coordinates of the focus
    \(F \left( {{\large\frac{p}{2}\normalsize}, 0} \right)\)

    Coordinates of the vertex
    \(M \left( {0,0} \right)\)

A parabola
  1. General equation of a parabola
    \(A{x^2} + Bxy + C{y^2} \) \(+\; Dx + Ey \) \(+\; F = 0,\)
    where \(B^2 – 4AC = 0\).
  2. General equation of a parabola, the axis of which is parallel to the \(y\)-axis
    \(A{x^2} + Dx + Ey \) \(+\; F = 0\;\) \(\left( {A \ne 0, E \ne 0} \right) ,\)
    or in the equivalent form
    \(y = a{x^2} + bx + c,\;\) \(p = {\large\frac{1}{2a}\normalsize}\)

    Equation of the directrix
    \(y = {y_0} – {\large\frac{p}{2}\normalsize}\),
    where \(p\) is the focal parameter.

    Coordinates of the focus
    \(F\left( {{x_0},{y_0} + {\large\frac{p}{2}\normalsize}} \right)\)

    Coordinates of the vertex
    \({x_0} = – {\large\frac{b}{{2a}}\normalsize},\;\) \({y_0} = ax_0^2 + b{x_0} + c =\) \({\large\frac{{4ac – {b^2}}}{{4a}}\normalsize}\)

General equation of a parabola, the axis of which is parallel to the y-axis
  1. Equation of a parabola with the vertex at the origin and the axis parallel to the \(y\)-axis
    \(y = a{x^2},\;\) \(p = {\large\frac{1}{{2a}}\normalsize}\)

    Equation of the directrix
    \(y = – {\large\frac{p}{2}\normalsize}\),
    where \(p\) is the focal parameter of the parabola.

    Coordinates of the focus
    \(F \left( {0, {\large\frac{p}{2}\normalsize}} \right)\)

    Coordinates of the vertex
    \(M \left( {0,0} \right)\)

Equation of a parabola with the vertex at the origin and the axis parallel to the y-axis