Latus Rectum Calculator

Calculate the latus rectum of parabolas, ellipses, and hyperbolas with step-by-step solutions.

Select Conic Section

For y = ax², p = 1/(4a). For y² = 4px, p is the coefficient divided by 4.
If provided, p will be calculated as 1/(4|a|)

Result

Latus Rectum Length
--
units
Semi-latus rectum --
Eccentricity --
Focus distance (c) --

Step-by-Step Solution

Latus Rectum = 4p (Parabola)

Understanding the Latus Rectum

The latus rectum of a conic section is a chord that passes through the focus and is perpendicular to the major axis (or the axis of symmetry for a parabola). It is an important geometric property that characterizes the "width" of the conic at the focus. The length of the latus rectum determines how "open" or "narrow" the curve is near its focus.

Latus Rectum Formulas by Conic Section

Parabola

For a parabola y² = 4px or x² = 4py, the latus rectum has length 4p, where p is the distance from vertex to focus.

LR = 4p = 1/|a| (for y = ax²)

Ellipse

For an ellipse x²/a² + y²/b² = 1 (a > b), the latus rectum passes through each focus.

LR = 2b²/a

Hyperbola

For a hyperbola x²/a² - y²/b² = 1, the formula is the same form as the ellipse.

LR = 2b²/a

Semi-Latus Rectum

Half the latus rectum. Often denoted by l or ℓ. For all conics: ℓ = a(1 - e²) where e is eccentricity.

ℓ = LR / 2

Properties of the Latus Rectum

  • The latus rectum always passes through a focus and is perpendicular to the principal axis.
  • For a parabola, there is one latus rectum (one focus). For ellipses and hyperbolas, there are two (one for each focus).
  • The semi-latus rectum is the parameter used in the polar equation of conics: r = ℓ/(1 + e cos θ).
  • A larger latus rectum indicates a "wider" curve near the focus.
  • For a circle (eccentricity = 0), the latus rectum equals the diameter (2r).

Relationship to Eccentricity

The semi-latus rectum ℓ is related to eccentricity by the unified formula: ℓ = a(1 - e²) for ellipses, ℓ = a(e² - 1) for hyperbolas, and ℓ = 2p for parabolas. This relationship is fundamental in orbital mechanics where conic sections describe planetary orbits.

Applications

  • Orbital mechanics: Kepler's laws use the semi-latus rectum to describe planetary orbits.
  • Optics: The latus rectum helps determine the focal properties of parabolic reflectors.
  • Architecture: Understanding conic sections for designing arches, domes, and reflective surfaces.
  • Engineering: Satellite trajectory calculations and antenna design.