Latus Rectum Calculator

Understanding the Latus Rectum of Conic Sections

In geometry and analytic mathematics, the latus rectum (plural: latera recta) is a chord that passes through a focal point of a conic section and is oriented perpendicular to the major or transverse symmetry axis. The term comes from the Latin words latus meaning "side" and rectum meaning "straight" or "upright."

Whether analyzing a parabolic trajectory in orbital mechanics, designing an optical reflector, or examining planetary orbits using an Ellipse Calculator or circular cross-sections with a Circle Calculator, the latus rectum provides a fundamental geometric measure of the curve's opening width and curvature at its focal point.

Latus Rectum Formulas by Conic Section

The mathematical formula for the length of the latus rectum depends on the type of conic section under investigation:

1. Parabola

For a parabola in standard orientation with vertex at the origin and focal length \(a\), described by the equation \(y^2 = 4ax\) or \(x^2 = 4ay\):

• Focal length: \(a\)
• Length of latus rectum: \(LR = 4a\)
• Semi-latus rectum: \(l = 2a\)
• Endpoints of chord (for \(y^2 = 4ax\)): \((a, 2a)\) and \((a, -2a)\)
• Directrix: \(x = -a\)

If the parabola is given in polynomial form \(y = Ax^2 + Bx + C\), the leading coefficient relates to the focal parameter by \(|A| = \frac{1}{4a}\), which gives \(LR = \frac{1}{|A|}\).

2. Ellipse

For an ellipse with semi-major axis \(a\) and semi-minor axis \(b\) (\(a \ge b\)) with canonical equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\):

• Length of latus rectum: \(LR = \frac{2b^2}{a}\)
• Semi-latus rectum: \(l = \frac{b^2}{a} = a(1 - e^2)\)
• Linear eccentricity (focal distance): \(c = \sqrt{a^2 - b^2}\)
• Orbital eccentricity: \(e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}\)
• Endpoints: \((\pm c, \pm \frac{b^2}{a})\)

3. Hyperbola

For a hyperbola with transverse semi-axis \(a\) and conjugate semi-axis \(b\) described by \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\):

• Length of latus rectum: \(LR = \frac{2b^2}{a}\)
• Semi-latus rectum: \(l = \frac{b^2}{a} = a(e^2 - 1)\)
• Focal distance: \(c = \sqrt{a^2 + b^2}\)
• Eccentricity: \(e = \frac{c}{a} = \sqrt{1 + \frac{b^2}{a^2}} > 1\)

4. Circle

A circle is a special symmetric ellipse where the semi-major and semi-minor axes are both equal to the radius \(r\) (\(a = b = r\)) and eccentricity \(e = 0\):

• Length of latus rectum: \(LR = \frac{2r^2}{r} = 2r\) (the diameter of the circle)
• Semi-latus rectum: \(l = r\) (the radius of the circle)

Step-by-Step Example Calculation

Suppose an ellipse has a semi-major axis \(a = 5\text{ cm}\) and a semi-minor axis \(b = 3\text{ cm}\).

1. Compute \(b^2 = 3^2 = 9\).
2. Compute the latus rectum length: \(LR = \frac{2b^2}{a} = \frac{2 \times 9}{5} = \frac{18}{5} = 3.6\text{ cm}\).
3. Compute the semi-latus rectum: \(l = \frac{LR}{2} = 1.8\text{ cm}\).
4. Compute the focal distance: \(c = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4\text{ cm}\).
5. Determine eccentricity: \(e = \frac{c}{a} = \frac{4}{5} = 0.8\).