1. What is a Hyperbola?

A hyperbola is a type of conic section that forms when a plane intersects both nappes of a double cone. It consists of two disconnected curves called branches that are mirror images of each other.

2. Hyperbola Equation Explained

The standard form of a horizontal hyperbola equation is:

Where:

• \( (h,k) \) — Center of the hyperbola
• \( a \) — Distance from center to vertices along transverse axis
• \( b \) — Distance from center to co-vertices along conjugate axis

Key Features: The hyperbola opens left and right. For a vertical hyperbola, the y-term comes first with a positive sign.

3. Hyperbola Properties

Details: Hyperbolas have two foci, two vertices, and two asymptotes that define their "opening angle." The eccentricity (e) is always greater than 1 for hyperbolas.

4. Using the Calculator

Tips: Enter the center coordinates (h,k), semi-major axis (a), and semi-minor axis (b). The calculator will determine all key properties of the hyperbola.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between a hyperbola and an ellipse?
A: The hyperbola has a subtraction in its standard equation and two separate branches, while the ellipse has addition and forms a single closed curve.

Q2: How do you find the foci of a hyperbola?
A: For horizontal hyperbolas, foci are at (h±c,k) where c² = a² + b². For vertical hyperbolas, they're at (h,k±c).

Q3: What do the asymptotes represent?
A: The asymptotes are lines that the hyperbola approaches but never touches. They define the "opening angle" of the hyperbola.

Q4: Can a hyperbola be a function?
A: A complete hyperbola is not a function (fails vertical line test), but each individual branch can be considered a function.

Q5: What are real-world applications of hyperbolas?
A: Hyperbolas appear in navigation systems (LORAN), telescope and antenna designs, and the paths of certain celestial objects.