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Semilatus Rectum — Definition, Formula & Examples

The semilatus rectum is the distance from the focus of a conic section to the curve, measured along a line perpendicular to the major axis (or axis of symmetry) through that focus.

For a conic section with semi-major axis aa (or focal parameter) and eccentricity ee, the semilatus rectum ℓ\ell is half the length of the latus rectum — the chord through a focus that is perpendicular to the principal axis. It equals ℓ=a(1−e2)\ell = a(1 - e^2) for an ellipse, ℓ=a(e2−1)\ell = a(e^2 - 1) for a hyperbola, and ℓ=2p\ell = 2p for a parabola y2=4pxy^2 = 4px (where pp is the distance from vertex to focus).

Key Formula

ℓ=a(1−e2)\ell = a(1 - e^2)
Where:
  • ℓ\ell = Semilatus rectum (distance from focus to curve along the perpendicular)
  • aa = Semi-major axis of the ellipse (or semi-transverse axis of a hyperbola, using $\ell = a(e^2 - 1)$)
  • ee = Eccentricity of the conic section

How It Works

The semilatus rectum appears naturally in the polar equation of any conic section with a focus at the origin: r=ℓ1+ecos⁡θr = \frac{\ell}{1 + e\cos\theta}. Here ℓ\ell controls the overall size of the curve while ee controls its shape. When θ=90°\theta = 90°, you get r=ℓr = \ell directly, confirming the geometric meaning. This makes the semilatus rectum especially useful in orbital mechanics and polar-form problems.

Worked Example

Problem: Find the semilatus rectum of an ellipse with semi-major axis a=10a = 10 and eccentricity e=0.6e = 0.6.
Step 1: Write the formula for the semilatus rectum of an ellipse.
ℓ=a(1−e2)\ell = a(1 - e^2)
Step 2: Substitute a=10a = 10 and e=0.6e = 0.6.
ℓ=10(1−0.36)=10(0.64)=6.4\ell = 10(1 - 0.36) = 10(0.64) = 6.4
Answer: The semilatus rectum is ℓ=6.4\ell = 6.4.

Why It Matters

In physics and astronomy, planetary orbits are described by the polar conic equation r=ℓ1+ecos⁡θr = \frac{\ell}{1 + e\cos\theta}, where ℓ\ell is the semilatus rectum. Knowing ℓ\ell lets you compute a planet's distance from the Sun at any angle without converting to Cartesian coordinates.

Common Mistakes

Mistake: Confusing the semilatus rectum with the full latus rectum.
Correction: The latus rectum is the entire chord through the focus; the semilatus rectum is half that length. If you compute a(1−e2)a(1-e^2), you already have the semi (half) value.