10.2 The Hyperbola

Hyperbola Fundamentals

A hyperbola is a conic section with two separate branches that extend infinitely in opposite directions. It's defined by a key distance property: for any point on the hyperbola, the difference of its distances to two fixed points (the foci) is constant. This definition drives everything else about how hyperbolas behave and how their equations work.

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Key Points of Hyperbolas

Every hyperbola has a set of defining features you need to know:

• Center: the midpoint between the two vertices
• Vertices: the two points where the hyperbola crosses its transverse axis (the axis that passes through both foci)
• Foci: two fixed points along the transverse axis, each farther from the center than the vertices. For any point on the hyperbola, the absolute difference of distances to the two foci equals $2a$
• Transverse axis: the segment connecting the vertices, with length $2a$
• Conjugate axis: perpendicular to the transverse axis through the center, with length $2b$. Its endpoints are called co-vertices
• Asymptotes: straight lines through the center that the branches approach but never touch

The relationship between $a$, $b$, and $c$ (the distance from center to each focus) is:

$c^2 = a^2 + b^2$

This is different from ellipses, where you subtract. For hyperbolas, $c > a$ always, which makes sense because the foci sit outside the vertices.

Standard-Form Equations for Hyperbolas

Which variable comes first tells you the orientation:

• Horizontal transverse axis (opens left and right): $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$
• Vertical transverse axis (opens up and down): $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$

Here, $(h, k)$ is the center, $a$ is the distance from center to each vertex along the transverse axis, and $b$ is the distance from center to each co-vertex along the conjugate axis.

A quick way to remember: the positive fraction always corresponds to the transverse axis direction. If $x$ is positive, the hyperbola opens horizontally. If $y$ is positive, it opens vertically.

The asymptote equations depend on orientation:

• Horizontal: $y - k = \pm \frac{b}{a}(x - h)$
• Vertical: $y - k = \pm \frac{a}{b}(x - h)$

Hyperbolas Centered at the Origin

When the center is $(0, 0)$, the equations simplify:

• Horizontal: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
• Vertical: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$

Steps to sketch a hyperbola centered at the origin:

1. Identify the transverse axis direction by checking which variable has the positive term
2. Plot the vertices: $(\pm a, 0)$ for horizontal, or $(0, \pm a)$ for vertical
3. Plot the co-vertices: $(0, \pm b)$ for horizontal, or $(\pm b, 0)$ for vertical
4. Draw the central rectangle using the vertices and co-vertices as guides (dimensions $2a$ by $2b$)
5. Draw the asymptotes as diagonals through the corners of that rectangle
6. Sketch the two branches passing through the vertices, curving toward (but never reaching) the asymptotes

The central rectangle and asymptotes are your best tools for getting an accurate sketch. Don't skip them.

Advanced Hyperbola Concepts

Non-Origin Centered Hyperbolas

When the center is at $(h, k)$ instead of the origin, the hyperbola is simply shifted. Every feature moves with it.

Steps to graph a non-origin centered hyperbola:

1. Plot the center $(h, k)$
2. Determine the transverse axis direction (positive term tells you) and read off $a$ and $b$
3. From the center, move $\pm a$ along the transverse axis to plot the vertices
4. From the center, move $\pm b$ along the conjugate axis to plot the co-vertices
5. Draw the central rectangle and its diagonals to get the asymptotes
6. Find the foci using $c^2 = a^2 + b^2$, then plot them at distance $c$ from the center along the transverse axis
7. Sketch the branches through the vertices, approaching the asymptotes

Example: Given $\frac{(x-3)^2}{16} - \frac{(y+1)^2}{9} = 1$, the center is $(3, -1)$, $a = 4$, $b = 3$, and the transverse axis is horizontal. Vertices are at $(7, -1)$ and $(-1, -1)$. The foci are at $(3 \pm 5, -1)$ since $c = \sqrt{16 + 9} = 5$.

Real-World Applications of Hyperbolas

Hyperbolas show up in several practical contexts:

• Navigation (TDOA): Systems like LORAN locate objects by measuring the time difference of signal arrival from two stations. Each pair of stations produces a hyperbola of possible positions, and the intersection of two hyperbolas gives the exact location.
• Orbital paths: Some comets follow hyperbolic trajectories around the sun, meaning they pass through the solar system once and never return.
• Reflective properties: Hyperbolic mirrors reflect light directed toward one focus so that it appears to come from the other focus. This principle is used in telescope designs (like the Cassegrain reflector).

Steps to solve real-world hyperbola problems:

1. Identify the given information and what you need to find
2. Set up a coordinate system (often placing the center at the origin)
3. Determine which standard form applies based on the geometry of the situation
4. Substitute known values to find $a$, $b$, or $c$
5. Solve for the unknown and interpret your answer in context

Additional Hyperbola Properties

• Eccentricity $e = \frac{c}{a}$, where $e > 1$ for all hyperbolas. Values close to 1 mean the branches are narrow and close to the asymptotes; larger values mean the branches are wider and more "open."
• Latus rectum: a chord through a focus, perpendicular to the transverse axis. Its length is $\frac{2b^2}{a}$. This helps you plot additional points on the curve for a more accurate graph.
• Directrices: two lines perpendicular to the transverse axis at distance $\frac{a}{e}$ from the center. For any point on the hyperbola, the ratio of its distance to a focus over its distance to the corresponding directrix equals $e$.