A **hyperbola** is a type of conic section that is formed by intersecting a double cone with a plane in such a way that the plane cuts through both halves of the cone. It is defined as the set of all points \( (x, y) \) in a plane such that the absolute difference of the distances from two fixed points, called **foci** (singular: focus), is constant.

### Key Characteristics of a Hyperbola:
1. **Foci**: Two fixed points inside each branch of the hyperbola.
2. **Vertices**: The points where the hyperbola intersects its principal axis.
3. **Asymptotes**: Two lines that the hyperbola approaches but never intersects. These lines define the general shape and orientation of the hyperbola.
4. **Axes**:
   - **Transverse Axis**: The line segment that passes through the vertices.
   - **Conjugate Axis**: The line segment perpendicular to the transverse axis and through the center of the hyperbola.

### Standard Equation of a Hyperbola:
1. **Horizontal Hyperbola** (opening left and right):
   \[
   \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
   \]
2. **Vertical Hyperbola** (opening up and down):
   \[
   \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1
   \]
   - \( (h, k) \) is the center of the hyperbola.
   - \( a \) is the distance from the center to each vertex.
   - \( b \) is related to the distance between the asymptotes.

### Example of a Horizontal Hyperbola:
Given the equation \(\frac{x^2}{9} - \frac{y^2}{16} = 1\), it represents a hyperbola with:
- Center at \( (0, 0) \)
- Vertices at \( (3, 0) \) and \( (-3, 0) \)
- Asymptotes with slopes \( \pm \frac{4}{3} \) (since \( \frac{b}{a} = \frac{4}{3} \)).

Would you like more details on hyperbolas, or do you have any specific questions?

### Related Questions:
1. How does the equation of a hyperbola differ from that of a parabola?
2. What is the significance of the foci in a hyperbola?
3. How can you derive the asymptotes of a hyperbola from its equation?
4. Can you explain the difference between a hyperbola and an ellipse?
5. How is the hyperbola used in real-world applications?

**Tip:** To graph a hyperbola, start by plotting the center, vertices, and asymptotes, then sketch the curves approaching the asymptotes.

Explore the definition of hyperbolas, their equations, and examples. Learn about the key characteristics such as foci, vertices, and asymptotes. Example equations and their graphical representation included.

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