Equations of Hyperbolas in Standard Form

 

Hyperbolas were introduced in two prior lessons:

In this current section, a hyperbola is positioned with its center at the origin and foci on the $x$-axis or $y$-axis.
These placements result in the so-called ‘standard forms’ for a hyperbola, which—as you'll see—are very simple equations!

If you're short on time, jump right to the summary of equations of hyperbolas.

Review and Additional Notation

Consider a hyperbola with foci $\,F_1\,$ and $\,F_2\,$ and hyperbola constant $\,k\,$.

Recall from the previous section that:
  • By the definition of hyperbola, every point $\,P\,$ on the hyperbola satisfies: $$ |d(P,F_1) - d(P,F_2)| = k$$ Here, $\,d(A,B)\,$ denotes the distance between points $\,A\,$ and $\,B\,$.
  • The major axis of a hyperbola is the line through the foci, shown at right dashed green.
  • The vertices of a hyperbola are the two points where the hyperbola intersects its major axis. At right, they are denoted by $\,V_1\,$ and $\,V_2\,$ (with $\,V_1\,$ closest to $\,F_1\,$).
  • A hyperbola ‘shows’ its hyperbola constant as the distance between the vertices.
    That is, $\,d(V_1,V_2) = k\,$.
Here is some additional notation:
  • Center of a Hyperbola:
    Let $\,C\,$ denote the midpoint of the line segment from $\,F_1\,$ to $\,F_2\,$.
    By definition, $\,C\,$ is the center of the hyperbola, and is shown at right in red.
  • Minor Axis of a Hyperbola:
    The minor axis of a hyperbola is the perpendicular bisector of the line segment between the foci. Equivalently, the minor axis is the line through the center that is perpendicular to the major axis. The minor axis is shown at right dashed purple.

The major axis of the hyperbola is the line through the foci; it is dashed green.

The center of the hyperbola is the midpoint of the line segment connecting the foci; it is shown in red.

The minor axis of the hyperbola is the line through the center that is perpendicular to the major axis;
it is dashed purple.

Derivation of the Equation of a Hyperbola:
Center at the Origin, Foci on the $x$-axis

  • Position the hyperbola:
    As shown at right, position a hyperbola with center at the origin
    and foci (marked with ‘x’) on the $x$-axis.
    With this positioning, the $x$-axis is the major axis; the $y$-axis is the minor axis.
  • Notation ($\,c\,$ and $\,a\,$):
    Define:
    • $c := \,$ the distance from the center (the origin) to each focus.
      Since $\,c\,$ is a distance, $\,c > 0\,$.
      The coordinates of the foci are therefore $\,(c,0)\,$ and $\,(-c,0)\,$.
    • $a := \,$ the distance from the center to each vertex.
      Since $\,a\,$ is a distance, $\,a > 0\,$; also, $\,a < c\,$.
      The coordinates of the vertices are therefore $\,(a,0)\,$ and $\,(-a,0)\,$.
    (Recall that ‘$\,:=\,$’ means ‘equal, by definition’.)
  • Hyperbola constant:
    The hyperbola constant is the distance between the vertices, which is $\,2a\,$.
Hyperbola with foci on $x$-axis
and center at the origin

$c := \,$ distance from center to each focus
$a := \,$ distance from center to each vertex
Note that $\,0 < a < c\,$.

Derivation of the Equation of a Hyperbola:
Center at the Origin, Foci on the $y$-axis

The derivation of the equation of a hyperbola with center at the origin and foci on the $y$-axis is nearly identical to the derivation above.

Summary:
Equations of hyperbolas with center at the origin
and foci on the $x$-axis or $y$-axis

In both cases:
$0 < a < c$
With the equations in standard form, the number $\,a^2\,$ is the denominator of the positive term.
The hyperbola constant is $\,2a\,$.
The foci are determined by solving the equation $\,c^2 = a^2 + b^2\,$ for $\,c\,$.

Be careful!
The variables $\,a\,$, $\,b\,$ and $\,c\,$ are used for both ellipses and hyperbolas,
but their relationship is different!
For ellipses, $\,c^2 = a^2 \color{red}{-} b^2\,$.
For hyperbolas, $\,c^2 = a^2 \color{red}{+} b^2\,$.

Foci on the $x$-axis


Equation of Hyperbola: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ When the foci are on the $\color{red}{x}$-axis,
the $\,\color{red}{x^2}$-term is positive.
Coordinates of foci: $\,(-c,0)\,$ and $\,(c,0)\,$

Foci on the $y$-axis


Equation of Hyperbola: $$\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$$ When the foci are on the $\color{red}{y}$-axis,
the $\,\color{red}{y^2}$-term is positive.
Coordinates of foci: $\,(0,-c)\,$ and $\,(0,c)\,$

TIPS:

EXAMPLE: Finding the Equation of a Hyperbola

Find the equation of the following hyperbola:

Also, find the coordinates of the foci.

Solution:
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Graphing Hyperbolas


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37      
AVAILABLE MASTERED
IN PROGRESS

(MAX is 37; there are 37 different problem types.)
Want textboxes to type in your answers? Check here: