Hyperbola
A hyperbola is an open conic section consisting of two separate mirror-image curves (branches), defined as the locus of all points for which the absolute difference of distances to two fixed points (foci) is constant. It has eccentricity greater than 1, distinguishing it from the ellipse (e < 1) and parabola (e = 1). Hyperbolas model phenomena such as the paths of comets under gravitational repulsion, sonic booms, and certain navigational systems (LORAN).
Key Formula
x²/a² − y²/b² = 1
LaTeX: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
| Symbol | Meaning | Unit |
|---|---|---|
| a | distance from centre to vertex along transverse axis | length units |
| b | distance related to conjugate axis, b² = c² − a² | length units |
| c | distance from centre to each focus | length units |
| x | horizontal coordinate | length units |
| y | vertical coordinate | length units |
Worked Example
Problem
A hyperbola has equation x²/9 − y²/16 = 1. Find (a) the vertices, (b) the foci, and (c) the equations of the asymptotes.
Solution
Step 1: Identify a² = 9 and b² = 16, so a = 3 and b = 4. Step 2: Vertices lie on the transverse (x) axis at (±a, 0) = (±3, 0). Step 3: c² = a² + b² = 9 + 16 = 25, so c = 5. Foci at (±5, 0). Step 4: Asymptote slopes = ±b/a = ±4/3. Equations: y = (4/3)x and y = −(4/3)x.
Answer
Vertices: (±3, 0); Foci: (±5, 0); Asymptotes: y = ±(4/3)x.
Comparing Standard Hyperbola Orientations
| Property | Horizontal (x²/a² − y²/b² = 1) | Vertical (y²/a² − x²/b² = 1) |
|---|---|---|
| Transverse axis | Along x-axis | Along y-axis |
| Vertices | (±a, 0) | (0, ±a) |
| Foci | (±c, 0), c² = a² + b² | (0, ±c), c² = a² + b² |
| Asymptotes | y = ±(b/a)x | y = ±(a/b)x |
| Eccentricity | e = c/a > 1 | e = c/a > 1 |
Interactive Tools
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Related Terms
From Greek hyperbolē (ὑπερβολή), meaning "excess" or "throwing beyond", from hyper- ("over, beyond") + bolē ("throw"). Apollonius of Perga coined the term in Conics (c. 200 BC) because the square of the ordinate "exceeds" the area applied to the latus rectum.