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8.1.3. Hyperbolas

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:

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:

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

$$
\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1.
$$

The branches open left and right from the center.

Vertical transverse axis

$$
\frac{(y - k)^{2}}{a^{2}} - \frac{(x - h)^{2}}{b^{2}} = 1.
$$

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:

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:

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:

  1. Group $x$-terms together and $y$-terms together.
  2. Factor out coefficients of $x^{2}$ and $y^{2}$.
  3. Complete the square in $x$ and in $y$ separately.
  4. Move constants to the right side.
  5. 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:

  1. Identify the center $(h,k)$ from the shifted variables $(x-h)$ and $(y-k)$.
  2. Determine $a$ and $b$ from the denominators $a^{2}$ and $b^{2}$.
  3. Decide the orientation:
    • If the $x$-part is positive, it opens left/right.
    • If the $y$-part is positive, it opens up/down.
  4. Plot the center and the vertices:
    • Horizontal: $(h \pm a, k)$
    • Vertical: $(h, k \pm a)$
  5. Draw the rectangle with sides $2a$ by $2b$ centered at $(h,k)$, aligned with the axes.
  6. Draw the two diagonals of this rectangle; these are the asymptotes.
  7. 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:

More detailed physical and applied examples are usually explored in later courses once you are familiar with functions and modeling.

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