The hyperbola

A hyperbola is the set of all points whose distances from two foci differ by the same constant 2a (the bigger distance minus the smaller). It has two separate branches.

Its parts, with the centre at the origin:

• The transverse axis passes through the foci; the vertices are a from the centre.
• The foci are c from the centre, and this time c > a.
• b is defined by c² = a² + b²: note the plus sign, the opposite way round from the ellipse.

Standard forms:

• Transverse axis along the x-axis: x²/a² – y²/b² = 1, foci (±c, 0), vertices (±a, 0).
• Transverse axis along the y-axis: y²/a² – x²/b² = 1, foci (0, ±c).
• The positive term shows the transverse axis; a² is always under the positive term, even if it is the smaller number.

• Eccentricity e = c/a > 1, because c > a.
• Latus rectum length 2b²/a, the same formula as the ellipse.

Example: x²/16 – y²/9 = 1.

• a = 4, b = 3, c = √(16 + 9) = 5.
• Foci (±5, 0), vertices (±4, 0).
• e = 5/4, latus rectum 2 × 9/4 = 9/2.

Example: y²/9 – x²/27 = 1. The positive term is y², so the transverse axis is the y-axis: a = 3, b² = 27, c = √36 = 6. Foci (0, ±6), e = 2, latus rectum 2 × 27/3 = 18.

Rewrite first if needed: 4x² – 9y² = 36, divided by 36, is x²/9 – y²/4 = 1.

Finding the equation. Vertices (0, ±4) and foci (0, ±5): a = 4, c = 5, so b² = 25 – 16 = 9, and the axis is the y-axis: y²/16 – x²/9 = 1.

When a = b, as in x² – y² = 4, the hyperbola is called equilateral (or rectangular), and e = √2.

Side by side: