The Hyperbola
A hyperbola is the locus of points whose distances from two foci differ by a constant, with standard form x2/a2 - y2/b2 = 1 and eccentricity e > 1.
What is The Hyperbola?
A hyperbola is the locus of points whose distances from two foci differ by a constant, with standard form x2/a2 - y2/b2 = 1 and eccentricity e > 1.
Key formula / rule: b2 = a2 (e2 - 1) and e > 1 always.
Key points
- Compare the defining properties of the three standard conics.
- Compute the asymptotes and foci of a hyperbola.
Common exam trap
Carrying the ellipse relation b2 = a2(1 - e2) into hyperbola problems.
Definitions
- Term
The Hyperbola
- Meaning
A hyperbola is the locus of points whose distances from two foci differ by a constant, with standard form x2/a2 - y2/b2 = 1 and eccentricity e > 1.
- Term
The Hyperbola — explanation
- Meaning
The sign change from the ellipse produces two branches and a pair of asymptotes, and b2 = a2(e2 - 1) replaces the ellipse relation. Rectangular hyperbolas have e = √(2).
Learning objectives
Compare the defining properties of the three standard conics.
Compute the asymptotes and foci of a hyperbola.
Formulae
- Key point
b2 = a2 (e2 - 1) and e > 1 always.
- Key point
Asymptotes are y = +/- (b/a) x.
- Key point
The conjugate hyperbola swaps the roles of the two axes.
Prerequisites
MAT-U3-CS-T1-S1-C2
Common mistakes
Carrying the ellipse relation b2 = a2(1 - e2) into hyperbola problems.
Ignoring the second branch when finding intersections.
Keywords
Hyperbola
Practice preview
What is the length of the latus rectum for the hyperbola x^2/36 - y^2/64 = 1?…
medium
Find the equations of the asymptotes for the hyperbola 4x^2 - 9y^2 = 36.…
medium
Which of the following statements is INCORRECT regarding a rectangular hyperbola?…
hard