Latus Rectum
The latus rectum is the chord through the focus perpendicular to the axis. Parabola: LR=4p. Ellipse/Hyperbola: LR=2b²/a. It characterizes the 'width' of the conic at the focus.
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Parabola: LR=4p; p is distance from vertex to focus. Ellipse and hyperbola share formula LR=2b²/a. Latus rectum endpoints are equidistant from focus.
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Why: The latus rectum appears in conic section definitions and optical properties. For a parabola, it's the focal chord perpendicular to the axis. Used in telescope and antenna design.
How: Parabola: LR=4p where p=focal distance. Ellipse/Hyperbola: LR=2b²/a. Requires a,b (or p) for the conic. The latus rectum passes through the focus.
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Conic Section Parameters
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
Parabola: LR=4p; Ellipse/Hyperbola: LR=2b²/a.
— Conic Sections
Latus rectum passes through focus.
— Definition
Key Takeaways
- • Parabola: where p is the focal distance.
- • Ellipse: with semi-axes a and b.
- • Hyperbola: (same formula as ellipse).
- • The latus rectum is the chord through a focus perpendicular to the major axis.
- • Semi-latus rectum p = b²/a for ellipse/hyperbola; p = 2p for parabola.
Did You Know?
Planetary orbits use the semi-latus rectum in polar equations.
Parabolic mirrors and antenna designs use latus rectum in their geometry.
The term comes from Latin "straight side," studied by Apollonius of Perga.
A circle (a=b) has latus rectum = 2a, equal to its diameter.
Kepler used the semi-latus rectum in his laws of planetary motion.
The latus rectum is a special focal chord of length 2p (semi-latus).
Understanding Latus Rectum
The latus rectum is a chord through a focus perpendicular to the major axis. Its length characterizes the conic section.
Expert Tips
Parabola
Only need p (focal distance). LR = 4p. Simple and quick.
Ellipse
a ≥ b. For circle, a=b gives LR = 2a (diameter).
Hyperbola
Same formula as ellipse but c = √(a²+b²) for foci.
Semi-latus
Semi-latus rectum p = LR/2. Used in polar equations r = ep/(1+ecosθ).
Frequently Asked Questions
What is the latus rectum?
A chord through a focus perpendicular to the major axis. Its length is a key parameter of conic sections.
Parabola formula?
LR = 4p, where p is the distance from the vertex to the focus.
Ellipse formula?
LR = 2b²/a, where a is semi-major and b is semi-minor axis.
Hyperbola formula?
LR = 2b²/a, same as ellipse. Geometry differs (c² = a²+b²).
What is semi-latus rectum?
Half the latus rectum. For ellipse/hyperbola, p = b²/a.
Circle latus rectum?
When a=b, LR = 2b²/a = 2a, the diameter.
Why is it useful?
Appears in orbital mechanics, optics, and polar equations of conics.
How to Use This Calculator
- Select conic type: parabola, ellipse, or hyperbola.
- Parabola: enter focal distance p. Ellipse/Hyperbola: enter semi-axes a and b.
- Click "Calculate" to get the latus rectum.
Disclaimer: For ellipse, a must be ≥ b. Results are for standard forms centered at the origin.
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