Hyperbolas are conic sections defined as the set of all points where the difference of the distances to two fixed points (foci) is constant. When dealing with word problems, the key is to identify the center, vertices, and foci from the given information. Once you have these, you can determine the equation of the hyperbola. Remember to consider whether the hyperbola opens horizontally or vertically, as this affects the form of the equation. Standard forms are $(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1)$ for horizontal hyperbolas and $(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1)$ for vertical hyperbolas, where $(h, k)$ is the center.
Word problems often describe scenarios involving distances, such as the difference in arrival times of signals or the range of a LORAN (Long Range Navigation) system. By carefully extracting the relevant parameters from the problem statement, you can construct the hyperbola's equation and solve for unknown quantities.
Match the following terms with their definitions:
| Term | Definition |
|---|---|
| 1. Focus | A. The midpoint between the vertices |
| 2. Vertex | B. A line passing through the foci |
| 3. Center | C. A point on the hyperbola closest to the center |
| 4. Transverse Axis | D. The two fixed points used to define a hyperbola |
| 5. Conjugate Axis | E. The line segment through the center perpendicular to the transverse axis |
A hyperbola is defined as the set of all points where the __________ of the distances to two fixed points, called __________, is constant. The line passing through the foci is the __________ axis, and the midpoint between the foci is the __________. The standard form equation for a hyperbola opening horizontally is $(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1)$, where (h, k) represents the __________.
Explain how the orientation (horizontal or vertical) of a hyperbola affects its equation and how you can determine the orientation from a word problem.
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