equilateral hyperbola
Very LowTechnical/Mathematical
Definition
Meaning
A hyperbola whose asymptotes are perpendicular to each other.
In analytic geometry, a specific type of hyperbola of the form xy = k/2 (where k is a constant), also known as a rectangular hyperbola because its asymptotes form right angles. In a broader mathematical context, it is a hyperbola where the semi-major and semi-minor axes are equal.
Linguistics
Semantic Notes
This term is exclusively used within mathematics, specifically in analytic geometry and conic sections. It is synonymous with 'rectangular hyperbola' in standard usage. The 'equilateral' refers to the equality of the semi-axes, not to triangles.
Dialectal Variation
British vs American Usage
Differences
No significant differences in usage or definition between British and American mathematical terminology.
Connotations
Purely technical, with identical connotations in both varieties.
Frequency
Extremely rare outside advanced mathematics textbooks or courses. Frequency is identical in both regions.
Vocabulary
Collocations
Grammar
Valency Patterns
The [noun] forms/is an equilateral hyperbola.The curve is an equilateral hyperbola because [reason].Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “No idioms exist for this technical term.”
Usage
Context Usage
Business
Not used.
Academic
Used exclusively in advanced mathematics, particularly in geometry, calculus, and physics contexts dealing with conic sections.
Everyday
Not used.
Technical
Core usage domain. Appears in mathematical proofs, engineering calculations (e.g., hyperbolic navigation systems), and scientific modelling.
Examples
By Part of Speech
adjective
British English
- The student identified the conic section as an equilateral hyperbola.
- They derived the equilateral hyperbola equation from the general form.
American English
- The curve graphed was an equilateral hyperbola.
- He solved the problem involving an equilateral hyperbola.
Examples
By CEFR Level
- The graph displayed a curve the teacher called an equilateral hyperbola.
- In our maths class, we learned that xy = 12 represents an equilateral hyperbola.
- The physicist noted that the relationship between pressure and volume under isothermal conditions plots as an equilateral hyperbola.
- To prove the curve was an equilateral hyperbola, we demonstrated that its asymptotes were orthogonal.
Learning
Memory Aids
Mnemonic
Think: 'Equal Lateral' → equal axes → the arms of the hyperbola stretch out equally from the centre, and its box is a square (making it rectangular).
Conceptual Metaphor
A symmetrical, open curve that mirrors itself perfectly across its perpendicular guides (asymptotes).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid literal translation that might imply a triangle ('равносторонний'). The standard Russian mathematical term is 'равносторонняя гипербола' or 'прямоугольная гипербола'. Stick with 'прямоугольная гипербола' as the primary equivalent.
Common Mistakes
- Confusing it with an equilateral triangle. Mispronouncing 'hyperbola' as 'hyperbole' (the rhetorical device). Incorrectly stating its asymptotes are 'parallel' instead of 'perpendicular'.
Practice
Quiz
What is another common name for an equilateral hyperbola?
FAQ
Frequently Asked Questions
Yes, in standard mathematical terminology, 'equilateral hyperbola' and 'rectangular hyperbola' are synonyms.
They appear in physics (e.g., Boyle's Law graphs), economics (certain demand curves), and engineering (in hyperbolic navigation systems like LORAN).
The term refers to the equality (equi-) of the lengths of the semi-major and semi-minor axes (lateral), making the associated 'auxiliary rectangle' a square.
The simplest Cartesian equation is xy = c, where c is a non-zero constant, centred at the origin with axes rotated 45 degrees.