Latus Rectum Calculator
Enter the parameters of your conic section to find the length of its latus rectum and the coordinates of the endpoints. The latus rectum is the chord drawn through a focus perpendicular to the major axis. This calculator handles all three conic types: parabola (from standard quadratic coefficients), ellipse, and hyperbola. Step-by-step working shows every calculation so you can follow along.
Formula
Worked example
For the parabola y = 2x² - 4x + 6: p = 1/(4 × 2) = 0.125, so LR = 4 × 0.125 = 0.5. The vertex is at h = -(-4)/(2 × 2) = 1, k = 6 - 16/8 = 4. The focus is at (1, 4.125). Latus rectum endpoints: (0.75, 4.125) and (1.25, 4.125).
What is the latus rectum?
The latus rectum of a conic section is a chord drawn through a focus and perpendicular to the principal axis (also called the major axis or axis of symmetry). The name comes from Latin, meaning "right side." For a parabola, there is exactly one latus rectum, because a parabola has only one focus. Ellipses and hyperbolas each have two foci, so they have two latus recta - one through each focus. The length of the latus rectum is a key geometric property that describes how wide or narrow the conic section is at its focus. A longer latus rectum means the curve is broader at the focal point; a shorter one means it is narrower and more tightly curved.
Latus rectum formulas for each conic section
For a parabola in standard quadratic form y = Ax² + Bx + C, the latus rectum length is LR = 1/|A|. The parameter p = 1/(4A) is the signed distance from the vertex to the focus: positive A means the parabola opens upward, negative A means it opens downward. The vertex is at (h, k) = (-B/(2A), C - B²/(4A)), and the focus sits at (h, k + p). The endpoints of the latus rectum are (h - LR/2, k + p) and (h + LR/2, k + p). For an ellipse given by x²/a² + y²/b² = 1 with a > b, the focal distance is c = sqrt(a² - b²) and the latus rectum through the right focus is LR = 2b²/a, with endpoints (c, b²/a) and (c, -b²/a). For a hyperbola x²/a² - y²/b² = 1, the focal distance is c = sqrt(a² + b²) and the same formula LR = 2b²/a applies, with endpoints (c, b²/a) and (c, -b²/a).
How to use this calculator
Select your conic section type at the top: parabola, ellipse, or hyperbola. For a parabola, enter the coefficients A, B, and C from the standard form y = Ax² + Bx + C. The coefficient A must be non-zero. For an ellipse, enter the semi-major axis a and semi-minor axis b, where a must be larger than b. For a hyperbola, enter the semi-transverse axis a and the semi-conjugate axis b - both must be positive. The calculator instantly returns the latus rectum length, the semi-latus rectum (half the length), the coordinates of the endpoints, and the eccentricity. The step-by-step panel shows every intermediate calculation so you can verify the working by hand.
Eccentricity and how it relates to the latus rectum
Eccentricity (e) is a single number that classifies any conic section: a circle has e = 0, an ellipse has 0 < e < 1, a parabola has e = 1, and a hyperbola has e > 1. For an ellipse, e = c/a where c = sqrt(a² - b²). For a hyperbola, e = c/a where c = sqrt(a² + b²). As the eccentricity of an ellipse approaches 1, the ellipse becomes very elongated and the latus rectum becomes very short relative to the major axis. As eccentricity approaches 0, the ellipse approaches a circle and the latus rectum approaches the diameter. For a hyperbola, larger eccentricity corresponds to a wider-opening curve.
Latus rectum formulas by conic section
| Conic | Standard form | Focus location | Latus rectum (LR) | Eccentricity |
|---|---|---|---|---|
| Parabola | y = Ax² + Bx + C | (h, k + p), p = 1/(4A) | LR = 4|p| = 1/|A| | e = 1 |
| Ellipse | x²/a² + y²/b² = 1, a > b | (±c, 0), c = sqrt(a²-b²) | LR = 2b²/a | 0 < e < 1 |
| Hyperbola | x²/a² - y²/b² = 1 | (±c, 0), c = sqrt(a²+b²) | LR = 2b²/a | e > 1 |
Standard-form equations and corresponding latus rectum formulas for horizontal orientations.
Frequently asked questions
What is the formula for the latus rectum of a parabola?
For a parabola in the form y = Ax² + Bx + C, the latus rectum length is LR = 1/|A|. Equivalently, LR = 4|p| where p = 1/(4A) is the distance from the vertex to the focus. For example, y = 2x² has A = 2, so LR = 1/2 = 0.5. A parabola in the form y² = 4ax has LR = 4a directly.
What is the formula for the latus rectum of an ellipse?
For an ellipse x²/a² + y²/b² = 1 with semi-major axis a and semi-minor axis b (where a > b), the latus rectum length is LR = 2b²/a. The endpoints of the latus rectum through the right focus (c, 0) are at (c, b²/a) and (c, -b²/a), where c = sqrt(a² - b²).
What is the formula for the latus rectum of a hyperbola?
For a hyperbola x²/a² - y²/b² = 1, the latus rectum formula is the same as for an ellipse: LR = 2b²/a. The difference is that the focal distance is c = sqrt(a² + b²) rather than sqrt(a² - b²). The endpoints through the right focus (c, 0) are at (c, b²/a) and (c, -b²/a).
How many latus recta does each conic section have?
A parabola has one latus rectum, because it has only one focus. An ellipse has two latus recta, one through each of its two foci. A hyperbola also has two latus recta, one through each focus. Each latus rectum is perpendicular to the principal axis and passes through a focus.
What is the semi-latus rectum?
The semi-latus rectum is half the length of the latus rectum. It equals the distance from the focus to the conic along the perpendicular to the axis. In orbital mechanics, the semi-latus rectum (often written as l or p) appears in the orbit equation r = l / (1 + e cos(theta)), making it a key parameter for describing planetary and satellite orbits.
Why does the latus rectum matter in conic sections?
The latus rectum gives a direct measure of how wide a conic section is at its focal point. In optics and engineering, parabolic reflectors and lenses use the focus heavily, so the latus rectum describes the width of the reflective or refractive surface at the focal plane. In orbital mechanics, the semi-latus rectum parameterizes the shape of elliptical, parabolic, and hyperbolic orbits in a unified way.