The ellipse

An ellipse is the set of all points whose distances from two fixed points (the foci) add up to the same constant, called 2a. You can draw one with a loop of string around two pins.

Names for the parts, with the centre at the origin:

• The major axis passes through both foci; its half-length is a. The minor axis is perpendicular to it; its half-length is b, with a > b.
• Each focus is a distance c from the centre.
• The end of the minor axis is a from each focus, so by Pythagoras a² = b² + c².

Standard forms:

• Major axis along the x-axis: x²/a² + y²/b² = 1, foci (±c, 0), vertices (±a, 0).
• Major axis along the y-axis: x²/b² + y²/a² = 1, foci (0, ±c).
• The larger denominator is a² and sits under the variable whose axis is the major axis.

Two more numbers:

• Eccentricity e = c/a. Since c < a, e < 1. The closer e is to 0, the rounder the ellipse; e = 0 would be a circle.
• Latus rectum (chord through a focus, perpendicular to the major axis): length 2b²/a.

Example: x²/100 + y²/36 = 1.

• a = 10, b = 6, c = √(100 – 36) = 8.
• Foci (±8, 0), vertices (±10, 0), major axis length 20, minor 12.
• e = 8/10 = 0.8, latus rectum 2 × 36/10 = 7.2.

Example: x²/16 + y²/25 = 1. The bigger number 25 is under y², so the major axis is the y-axis: a = 5, b = 4, c = 3, foci (0, ±3).

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

Finding the equation.

• Foci (±3, 0) and vertices (±5, 0): c = 3, a = 5, so b² = 25 – 9 = 16: x²/25 + y²/16 = 1.
• Centre at the origin, major axis on the x-axis, passing through (4, 0) and (2, 3): (4, 0) gives a = 4; then 4/16 + 9/b² = 1, so b² = 12: x²/16 + y²/12 = 1.