Latus rectum of an ellipse

A latus rectum of an ellipse is a chord through a focus perpendicular to the major axis, with its ends on the ellipse. An ellipse has two of them, one through each focus.

Its upper end A = (c, l) = (ae, l) lies on x²/a² + y²/b² = 1, so l² = b²(1 − e²). Since 1 − e² = b²/a², l = b²/a, and by symmetry the full latus rectum is 2b²/a.

For x²/25 + y²/9 = 1 the latus rectum is 2 × 9/5 = 18/5.

Equation from vertices (±13, 0) and foci (±5, 0): a = 13, c = 5, b² = 169 − 25 = 144, giving x²/169 + y²/144 = 1.

Equation from a major axis of length 20 and foci (0, ±5): the major axis is the y-axis, a = 10, b² = 100 − 25 = 75, giving x²/75 + y²/100 = 1.

Equation from two points: if the major axis is along the x-axis and the ellipse passes through (4, 3) and (−1, 4), then 16/a² + 9/b² = 1 and 1/a² + 16/b² = 1. Solving gives a² = 247/7 and b² = 247/15, so 7x² + 15y² = 247.