Table of Contents
Standard forms of hyperbolas
In analytic geometry, a hyperbola is defined as a conic section and also as the set of points whose distances to two fixed points (the foci) have a constant difference. In coordinates this leads to simple “standard form” equations.
We will always assume $a>0$ and $b>0$ in what follows.
Hyperbolas centered at the origin
A hyperbola with center at the origin $(0,0)$ has one of two basic orientations.
Horizontal transverse axis
This hyperbola opens left and right:
$$
\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1.
$$
Key features:
- Center: $(0,0)$
- Vertices: $(\pm a, 0)$
- Transverse axis: horizontal line through the center (the line containing the two vertices)
- Conjugate axis: vertical line through the center
- Asymptotes: lines through the center that the branches approach:
$$
y = \pm \frac{b}{a} x.
$$
The graph has two branches: one opening to the right, one to the left.
Vertical transverse axis
This hyperbola opens up and down:
$$
\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1.
$$
Key features:
- Center: $(0,0)$
- Vertices: $(0,\pm a)$
- Transverse axis: vertical line through the center
- Conjugate axis: horizontal line through the center
- Asymptotes:
$$
y = \pm \frac{a}{b} x.
$$
Again, there are two branches: one opening upward, one downward.
The only difference between these two standard forms is the sign pattern and which variable comes first: the positive term determines the direction of opening.
Hyperbolas centered at $(h,k)$
Shifting a hyperbola away from the origin gives the general standard forms.
Horizontal transverse axis
$$
\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1.
$$
- Center: $(h,k)$
- Vertices: $(h \pm a,\, k)$
- Asymptotes:
$$
y - k = \pm \frac{b}{a} (x - h).
$$
The branches open left and right from the center.
Vertical transverse axis
$$
\frac{(y - k)^{2}}{a^{2}} - \frac{(x - h)^{2}}{b^{2}} = 1.
$$
- Center: $(h,k)$
- Vertices: $(h,\, k \pm a)$
- Asymptotes:
$$
y - k = \pm \frac{a}{b} (x - h).
$$
The branches open up and down from the center.
In both cases, the asymptotes pass through the center and give a “guiding box” for sketching.
Geometric parameters: $a$, $b$, $c$ and the foci
For hyperbolas there are three related parameters: $a$, $b$, and $c$. The parameters $a$ and $b$ appear directly in the equation; $c$ is the distance from the center to each focus.
For the horizontal hyperbola
$$
\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1
$$
the foci lie on the same horizontal line as the center:
- Foci: $(h \pm c,\, k)$
For the vertical hyperbola
$$
\frac{(y - k)^{2}}{a^{2}} - \frac{(x - h)^{2}}{b^{2}} = 1
$$
the foci lie on the vertical line through the center:
- Foci: $(h,\, k \pm c)$
The relation among $a$, $b$, and $c$ is characteristic for hyperbolas:
$$
c^{2} = a^{2} + b^{2}.
$$
This is different from the ellipse case, where $c^{2} = a^{2} - b^{2}$ (covered in that chapter). For hyperbolas, the foci are always farther from the center than the vertices, since $c > a$.
Asymptotes in more detail
Hyperbolas have two straight-line asymptotes that intersect at the center. The branches get closer and closer to these lines as $|x|$ or $|y|$ grows.
For a centered hyperbola
$$
\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1,
$$
you can see the asymptotes by setting the right-hand side to $0$ instead of $1$:
$$
\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 0
\quad\Rightarrow\quad
\frac{x^{2}}{a^{2}} = \frac{y^{2}}{b^{2}}
\quad\Rightarrow\quad
y = \pm \frac{b}{a} x.
$$
For a shifted hyperbola
$$
\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1,
$$
the same idea (and a change of variables $X = x-h$, $Y = y-k$) leads to
$$
y - k = \pm \frac{b}{a} (x - h).
$$
These equations are very useful for sketching: draw the center, then the asymptotes, then place the vertices; the branches of the hyperbola approach the asymptotes but never cross them.
Identifying the equation of a hyperbola
Given an equation, you can often tell it represents a hyperbola and put it into standard form.
A conic of the form
$$
Ax^{2} + Cy^{2} + Dx + Ey + F = 0
$$
(with no $xy$ term) is a hyperbola if $A$ and $C$ have opposite signs.
To convert to standard form when the axes are not rotated:
- Group $x$-terms together and $y$-terms together.
- Factor out coefficients of $x^{2}$ and $y^{2}$.
- Complete the square in $x$ and in $y$ separately.
- Move constants to the right side.
- Divide both sides so that the right side equals $1$.
The resulting expression will match one of the standard hyperbola forms, and from there you can read off the center $(h,k)$, $a^{2}$, and $b^{2}$.
Relationship to the distance definition
The classical definition of a hyperbola uses distances to two fixed points (the foci). A point $P$ lies on the hyperbola if
$$
|PF_{1} - PF_{2}| = 2a,
$$
where $F_{1}$ and $F_{2}$ are the foci, and $2a$ is a positive constant.
If the foci are at $(\pm c,0)$ and the center is at the origin, using this distance definition together with $c^{2} = a^{2} + b^{2}$ leads (through algebra) to the standard forms
$$
\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1
\quad\text{or}\quad
\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1,
$$
depending on the orientation. This ties the geometric distance property to the algebraic equation.
Sketching a hyperbola from its standard form
To sketch a hyperbola given in standard form:
- Identify the center $(h,k)$ from the shifted variables $(x-h)$ and $(y-k)$.
- Determine $a$ and $b$ from the denominators $a^{2}$ and $b^{2}$.
- Decide the orientation:
- If the $x$-part is positive, it opens left/right.
- If the $y$-part is positive, it opens up/down.
- Plot the center and the vertices:
- Horizontal: $(h \pm a, k)$
- Vertical: $(h, k \pm a)$
- Draw the rectangle with sides $2a$ by $2b$ centered at $(h,k)$, aligned with the axes.
- Draw the two diagonals of this rectangle; these are the asymptotes.
- Sketch the two branches approaching the asymptotes and passing through the vertices, opening in the correct direction.
This rectangle-and-asymptotes method provides a fast, visual way to construct an accurate graph.
Hyperbolas and rectangular hyperbolas
A special case occurs when $a = b$. For example,
$$
\frac{x^{2}}{a^{2}} - \frac{y^{2}}{a^{2}} = 1
\quad\Rightarrow\quad
x^{2} - y^{2} = a^{2}.
$$
Here the asymptotes are
$$
y = \pm x,
$$
which are perpendicular. When the asymptotes are perpendicular, the hyperbola is often called a rectangular hyperbola.
A very common example (though not in standard conic form) is
$$
xy = k,
$$
which also has perpendicular asymptotes and branches in opposite quadrants.
Basic applications and appearances
Hyperbolas arise naturally in various contexts:
- Difference of distances: The original geometric definition can be used to describe paths where the difference in distance to two points is constant (for example, certain navigation or signal problems).
- Certain mirrors and antennas: Curves related to hyperbolas appear in reflection problems where rays from one focus are related to another line or point.
- Equations with inverse relationships: Rectangular hyperbolas, such as $y = \dfrac{k}{x}$, model situations where one quantity is inversely proportional to another.
More detailed physical and applied examples are usually explored in later courses once you are familiar with functions and modeling.