Command Palette

Search for a command to run...

What is a Hyperbola?

Have you ever seen the shape of cooling towers at power plants? Or perhaps noticed the shadow of a flashlight on the wall forming an open curve? Well, shapes like these are examples of hyperbolas in real life!

A hyperbola is a curve formed when a plane cuts through a double cone at a certain angle. Unlike an ellipse which forms a closed curve, a hyperbola is actually formed from two separate curves that face each other.

Mathematically, a hyperbola is defined as the locus of points where the absolute difference of distances to two fixed points is always constant. These two fixed points are called the foci of the hyperbola. For every point PP on the hyperbola, the difference PF1PF2=2a|PF_1 - PF_2| = 2a (constant), whereas in an ellipse the sum of distances is constant: PF1+PF2=2aPF_1 + PF_2 = 2a.

Basic Concept of Hyperbola
Hyperbola with two foci showing constant difference in distances

Look at the visualization above! A hyperbola has two separate branches. For every point PP on the hyperbola, the difference in distance from PP to both foci F1F_1 and F2F_2 is always constant.

Components of a Hyperbola

Before diving into formulas, let's get acquainted with the important parts of a hyperbola. Each component has its own role in determining the shape and properties of the hyperbola.

Parts of a Hyperbola
Important components of a hyperbola with horizontal major axis.

The components of a hyperbola you need to know:

  1. Center of hyperbola is the midpoint between the two foci, usually denoted by OO.

  2. Foci (F1F_1 and F2F_2) are two fixed points that serve as the reference for the definition of a hyperbola. The distance between the two foci is called the focal distance.

  3. Vertices (A1A_1 and A2A_2) are the closest points between the two branches of the hyperbola. The line connecting the two vertices is called the major axis.

  4. Major axis is the line that passes through the center and both foci of the hyperbola.

  5. Asymptotes are lines that are approached by the branches of the hyperbola as they extend to infinity. The hyperbola never touches its asymptotes, but the farther from the center, the closer the curve approaches the asymptote lines.

The main difference between a hyperbola and an ellipse: a hyperbola has asymptotes and consists of two separate branches, while an ellipse is a closed curve without asymptotes.

Equation of a Hyperbola

There are several forms depending on orientation and center position. We will discuss two cases:

Center at the Origin

If the center of the hyperbola is at O(0,0)O(0,0), there are two possible orientations:

Horizontal Major Axis
Hyperbola x2a2y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 with horizontal major axis.

When the major axis is parallel to the XX axis (horizontal), the equation of the hyperbola is:

x2a2y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

If we visualize it, it will look like this:

Vertical Major Axis
Hyperbola y2a2x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 with vertical major axis.

When the major axis is parallel to the YY axis (vertical), the equation of the hyperbola is:

y2a2x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

Shifted Center

If the center of the hyperbola is not at the origin, but at point (h,k)(h, k), the equation becomes:

(xh)2a2(yk)2b2=1(horizontal major axis)\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \quad \text{(horizontal major axis)}
(yk)2a2(xh)2b2=1(vertical major axis)\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \quad \text{(vertical major axis)}

Important Properties of Hyperbola

There are several mathematical relationships that always hold for every hyperbola:

c2=a2+b2c^2 = a^2 + b^2

where:

  • aa is the distance from center to vertex (semi-major axis)
  • bb is the constant that determines the shape of the hyperbola (semi-conjugate axis)
  • cc is the distance from center to focus

This relationship is different from an ellipse which uses c2=a2b2c^2 = a^2 - b^2.

Eccentricity of a hyperbola is defined as:

e=cae = \frac{c}{a}

The eccentricity value of a hyperbola is always e>1e > 1. This is because c>ac > a (from the relationship c2=a2+b2c^2 = a^2 + b^2, so c=a2+b2>ac = \sqrt{a^2 + b^2} > a). The larger the value of ee, the more "open" or wide the hyperbola becomes. For comparison: a circle has e=0e = 0, an ellipse has 0<e<10 < e < 1, a parabola has e=1e = 1, and a hyperbola has e>1e > 1.

Asymptote equations for a hyperbola with center at (0,0)(0,0):

y=±bax(horizontal major axis)y = \pm \frac{b}{a}x \quad \text{(horizontal major axis)}
y=±abx(vertical major axis)y = \pm \frac{a}{b}x \quad \text{(vertical major axis)}

Asymptotes are "guides" for the hyperbola branches. The farther from the center, the closer the hyperbola curve approaches the asymptote lines, but never touches them.

Exercises

  1. Determine the equation of a hyperbola with center at (0,0)(0,0), vertices at (±2,0)(\pm 2, 0), and foci at (±3,0)(\pm 3, 0).

  2. Given the hyperbola x225y29=1\frac{x^2}{25} - \frac{y^2}{9} = 1. Determine the coordinates of the foci, eccentricity, and asymptote equations.

  3. A hyperbola has its center at (1,2)(1, -2), horizontal major axis, a=3a = 3, and b=4b = 4. Determine the equation of the hyperbola.

  4. Determine the asymptote equations of the hyperbola (y+1)24(x2)29=1\frac{(y+1)^2}{4} - \frac{(x-2)^2}{9} = 1.

Answer Key

  1. Solution:

    Given:

    • Center at (0,0)(0,0)
    • Vertices at (±2,0)(\pm 2, 0), so a=2a = 2
    • Foci at (±3,0)(\pm 3, 0), so c=3c = 3
    • Horizontal major axis (since vertices and foci are on the XX axis)

    Use the relationship c2=a2+b2c^2 = a^2 + b^2:

    32=22+b23^2 = 2^2 + b^2
    9=4+b29 = 4 + b^2
    b2=5b^2 = 5

    Equation of hyperbola with horizontal major axis:

    x24y25=1\frac{x^2}{4} - \frac{y^2}{5} = 1
  2. Solution:

    From the equation x225y29=1\frac{x^2}{25} - \frac{y^2}{9} = 1:

    • a2=25a^2 = 25, so a=5a = 5
    • b2=9b^2 = 9, so b=3b = 3

    Since it's in the form x2a2y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, the major axis is horizontal.

    Calculate cc:

    c2=a2+b2=25+9=34c^2 = a^2 + b^2 = 25 + 9 = 34
    c=345,831c = \sqrt{34} \approx 5{,}831

    Coordinates of foci: (±34,0)(\pm \sqrt{34}, 0)

    Eccentricity:

    e=ca=3455,83151,166e = \frac{c}{a} = \frac{\sqrt{34}}{5} \approx \frac{5{,}831}{5} \approx 1{,}166

    Asymptote equations (horizontal major axis):

    y=±bax=±35xy = \pm \frac{b}{a}x = \pm \frac{3}{5}x
  3. Solution:

    Given:

    • Center: (h,k)=(1,2)(h,k) = (1, -2)
    • Horizontal major axis
    • a=3a = 3, so a2=9a^2 = 9
    • b=4b = 4, so b2=16b^2 = 16

    Equation of hyperbola with center (h,k)(h,k) and horizontal major axis:

    (xh)2a2(yk)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1

    Substituting values:

    (x1)29(y+2)216=1\frac{(x-1)^2}{9} - \frac{(y+2)^2}{16} = 1
  4. Solution:

    From the equation (y+1)24(x2)29=1\frac{(y+1)^2}{4} - \frac{(x-2)^2}{9} = 1:

    • Center: (h,k)=(2,1)(h,k) = (2, -1)
    • a2=4a^2 = 4, so a=2a = 2
    • b2=9b^2 = 9, so b=3b = 3
    • Vertical major axis (since it's in the form (yk)2a2(xh)2b2=1\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1)

    For a hyperbola with center (h,k)(h,k) and vertical major axis, the asymptote equations:

    yk=±ab(xh)y - k = \pm \frac{a}{b}(x - h)

    Substituting values:

    y(1)=±23(x2)y - (-1) = \pm \frac{2}{3}(x - 2)
    y+1=±23(x2)y + 1 = \pm \frac{2}{3}(x - 2)
    y=1±23(x2)y = -1 \pm \frac{2}{3}(x - 2)

    So the asymptote equations are:

    y=1+23(x2)y = -1 + \frac{2}{3}(x - 2)
    y=123(x2)y = -1 - \frac{2}{3}(x - 2)