Hyperbolas and Their Equations

A hyperbola is a type of conic section. Geometrically, it is the set of all points in a plane such that the absolute difference of their distances to two fixed points (called the foci) is a constant. Unlike an ellipse, which is a closed curve, a hyperbola consists of two separate, mirror-image curves called branches.

Standard Equations of Hyperbolas

The standard equation of a hyperbola depends on whether its branches open horizontally (left and right) or vertically (up and down). Assuming the center is at the origin $(0,0)$:

1. Horizontal Hyperbola (Opens Left/Right) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

• The transverse axis (the line passing through the vertices and foci) is on the x-axis.
• Vertices: $(\pm a, 0)$
• Foci: $(\pm c, 0)$
• Asymptotes: $y = \pm \frac{b}{a}x$

2. Vertical Hyperbola (Opens Up/Down) $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$

• The transverse axis is on the y-axis.
• Vertices: $(0, \pm a)$
• Foci: $(0, \pm c)$
• Asymptotes: $y = \pm \frac{a}{b}x$

For both types of hyperbolas, the relationship between $a$, $b$, and $c$ is given by the Pythagorean-like equation: $c^2 = a^2 + b^2$ where $c$ is the distance from the center to the foci.

Key Features

• Transverse Axis: The segment connecting the two vertices. Its length is $2a$.
• Conjugate Axis: The segment perpendicular to the transverse axis, passing through the center. Its length is $2b$.
• Asymptotes: Two intersecting lines that the branches of the hyperbola approach but never touch as they head toward infinity.

Example 1: Finding Asymptotes and Foci

Problem: Find the asymptotes and foci of the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$.

Solution:

1. Identify the orientation: Because the $x^2$ term is positive, this is a horizontal hyperbola.
2. Identify $a^2$ and $b^2$:
   • $a^2 = 16 \implies a = 4$
   • $b^2 = 9 \implies b = 3$
3. Find $c$ to locate the foci:
   • $c^2 = a^2 + b^2 = 16 + 9 = 25$
   • $c = 5$
   • The foci are at $(\pm 5, 0)$.
4. Find the asymptotes:
   • For a horizontal hyperbola, the formula is $y = \pm \frac{b}{a}x$.
   • The asymptotes are $y = \pm \frac{3}{4}x$.

Example 2: Writing the Equation

Problem: Write the equation of a hyperbola with vertices $(0, \pm 4)$ and foci $(0, \pm 5)$.

Solution:

1. Identify the orientation: The vertices and foci are on the y-axis, meaning this is a vertical hyperbola. The standard form is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$.
2. Find $a$ and $c$:
   • The distance from the center to a vertex is $a = 4$, so $a^2 = 16$.
   • The distance from the center to a focus is $c = 5$, so $c^2 = 25$.
3. Calculate $b^2$:
   • Using $c^2 = a^2 + b^2$, we get $25 = 16 + b^2$.
   • $b^2 = 25 - 16 = 9$.
4. Write the final equation: $\frac{y^2}{16} - \frac{x^2}{9} = 1$