How the Latus Rectum Works
The latus rectum of a conic section is the chord that passes through a focus, drawn perpendicular to the major/transverse axis (or the axis of symmetry for a parabola), with both endpoints on the curve. Its length is a fixed geometric property of the conic, determined entirely by the shape's defining parameters — it does not depend on where along the curve you measure.
Formula and method
For a parabola in standard form y² = 4ax (vertex at the origin, focus at (a, 0)), the latus rectum has length L = 4a, with endpoints (a, 2a) and (a, −2a). For an ellipse x²/a² + y²/b² = 1 with semi-major axis a and semi-minor axis b (a ≥ b), the latus rectum length is L = 2b²/a, giving a semi-latus rectum l = b²/a. The same length formula, L = 2b²/a, applies to a hyperbola x²/a² − y²/b² = 1, using the semi-transverse axis a and semi-conjugate axis b. Eccentricity is e = c/a for ellipses and hyperbolas, where c = √(a² − b²) for an ellipse and c = √(a² + b²) for a hyperbola; a parabola always has e = 1.
Choosing a and b
Enter a and b in the same length unit — inches, meters, or unitless graphing coordinates all work, as long as they match. For a parabola, only a is used: it is the distance from the vertex to the focus, and b is ignored. For an ellipse, a must be greater than or equal to b, since a always denotes the longer semi-major axis in the standard form used here; entering the smaller axis as a describes a different ellipse than intended. For a hyperbola, a and b are independent positive numbers with no ordering requirement.
Common sources of error
- Mixing up a and b: for an ellipse, entering the smaller axis as a produces an eccentricity greater than 1, which is impossible for a real ellipse — always put the larger axis in a.
- Full axis vs. semi-axis: a and b are half the major/minor (or transverse/conjugate) axis lengths, not the full axis lengths, so double-check which one your source material reports.
- Forgetting the factor of 2: the semi-latus rectum (l = b²/a) is half the full latus rectum length (L = 2b²/a) — confirm which one your problem is asking for.
Applications
The latus rectum is used to sketch conics accurately, since it fixes how "wide" the curve is at the focus; in optics and antenna design, where parabolic reflectors focus parallel rays to a point at distance a from the vertex; and in orbital mechanics, where the semi-latus rectum l = a(1 − e²) describes the size of an elliptical orbit independent of its orientation in space.