Latus Rectum Calculator

Calculate the latus rectum length for parabolas, ellipses, and hyperbolas

Enter the parameters of your conic section to find the length of its latus rectum. The latus rectum is the focal chord perpendicular to the major axis.

Examples

Click on any example to load it into the calculator

Standard Parabola

parabola

Basic parabola with parameter p = 2

p: 2

a: undefined

b: undefined

Ellipse Example

ellipse

Ellipse with semi-major axis 5 and semi-minor axis 3

p: undefined

a: 5

b: 3

Hyperbola Example

hyperbola

Hyperbola with semi-major axis 4 and semi-minor axis 2

p: undefined

a: 4

b: 2

Large Parabola

parabola

Parabola with larger parameter p = 10

p: 10

a: undefined

b: undefined

Other Titles
Understanding Latus Rectum Calculator: A Comprehensive Guide
Master the concept of latus rectum in conic sections and learn how to calculate it for parabolas, ellipses, and hyperbolas

What is Latus Rectum? Fundamental Concepts and Definition

  • Latus rectum is a focal chord perpendicular to the major axis
  • It passes through the focus of a conic section
  • Essential geometric property for understanding conic sections
The latus rectum is a fundamental geometric property of conic sections that represents the length of the chord passing through the focus and perpendicular to the major axis. This concept is crucial in analytical geometry and helps characterize the shape and properties of parabolas, ellipses, and hyperbolas.
For a parabola with equation y² = 4px, the latus rectum has length 4p, where p is the distance from the vertex to the focus. This relationship provides a direct connection between the parabola's opening width and its focal parameter.
In ellipses and hyperbolas, the latus rectum length is calculated as 2b²/a, where 'a' is the semi-major axis and 'b' is the semi-minor axis. This formula demonstrates how the eccentricity and shape of the conic affect the focal chord length.
The term 'latus rectum' comes from Latin, meaning 'straight side,' which aptly describes its geometric nature as a straight line segment through the focus perpendicular to the axis of symmetry.

Basic Latus Rectum Calculations

  • Parabola y² = 8x has p = 2, so latus rectum = 4(2) = 8
  • Ellipse x²/25 + y²/9 = 1 has a = 5, b = 3, so latus rectum = 2(9)/5 = 3.6
  • Hyperbola x²/16 - y²/4 = 1 has a = 4, b = 2, so latus rectum = 2(4)/4 = 2
  • Unit parabola y² = 4x has latus rectum = 4

Step-by-Step Guide to Using the Latus Rectum Calculator

  • Master the input methods for different conic sections
  • Understand parameter relationships and formulas
  • Interpret results and verify calculations
Our latus rectum calculator provides a streamlined interface for computing focal chord lengths across all major conic sections with precision and ease.
Parabola Calculations:
  • Parameter Input: Enter the value of p from the standard form y² = 4px or x² = 4py
  • Formula Applied: Latus rectum = 4p, where p is the focal parameter
  • Geometric Meaning: The chord through the focus perpendicular to the axis of symmetry
Ellipse Calculations:
  • Axis Inputs: Enter both semi-major axis (a) and semi-minor axis (b)
  • Formula Applied: Latus rectum = 2b²/a, relating the axes to focal chord length
  • Important Note: Ensure a > b for proper ellipse definition
Hyperbola Calculations:
  • Axis Inputs: Enter semi-major axis (a) and semi-minor axis (b)
  • Formula Applied: Latus rectum = 2b²/a, same as ellipse formula
  • Geometric Context: The chord through either focus perpendicular to the transverse axis

Calculator Usage Examples

  • Parabola: Input p = 3 → Latus rectum = 4(3) = 12 units
  • Ellipse: Input a = 6, b = 4 → Latus rectum = 2(16)/6 = 5.33 units
  • Hyperbola: Input a = 8, b = 6 → Latus rectum = 2(36)/8 = 9 units
  • Verification: Check that calculated values match geometric properties

Real-World Applications of Latus Rectum in Engineering and Science

  • Satellite orbital mechanics and trajectory analysis
  • Antenna and reflector design optimization
  • Optical systems and lens engineering
  • Architectural and structural applications
The latus rectum concept finds extensive applications across engineering, physics, and architectural design, where understanding conic section properties is crucial for optimal performance.
Satellite and Orbital Mechanics:
  • Orbital Analysis: Satellite orbits follow elliptical paths, and the latus rectum helps determine orbital characteristics and fuel requirements for trajectory corrections.
  • Communication Systems: Geostationary satellite positioning relies on elliptical orbit calculations where latus rectum affects signal coverage patterns.
Antenna and Reflector Design:
  • Parabolic Antennas: The latus rectum determines the focal length-to-diameter ratio, affecting gain and beam width characteristics.
  • Radio Telescopes: Large parabolic reflectors use latus rectum calculations to optimize sensitivity and directional accuracy.
Optical Systems Engineering:
  • Camera Lenses: Elliptical and parabolic lens elements require precise latus rectum calculations for aberration correction and image quality.
  • Laser Systems: Beam shaping and focusing systems utilize conic section properties for precise energy distribution.
Architectural Applications:
  • Arch Design: Parabolic and elliptical arches use latus rectum calculations for structural optimization and aesthetic proportions.
  • Acoustic Design: Concert halls and auditoriums employ elliptical geometries where latus rectum affects sound reflection and distribution.

Engineering Applications

  • GPS satellite orbit: Ellipse with a = 26,560 km, latus rectum calculation for positioning accuracy
  • Parabolic dish antenna: p = 0.25m diameter dish, latus rectum = 1m affects focal point placement
  • Architectural arch: Parabolic bridge span with p = 50m, latus rectum = 200m for structural analysis
  • Telescope mirror: f/4 parabolic primary with calculated latus rectum for optimal focus

Common Misconceptions and Correct Methods in Latus Rectum Calculations

  • Avoiding formula confusion between conic types
  • Understanding parameter relationships correctly
  • Preventing geometric interpretation errors
Understanding latus rectum requires careful attention to geometric definitions and formula applications, as common misconceptions can lead to significant calculation errors.
Formula Confusion Prevention:
  • Parabola vs. Ellipse: Remember that parabolas use 4p while ellipses and hyperbolas use 2b²/a. These are fundamentally different relationships.
  • Parameter Identification: Ensure correct identification of 'p' in parabolas (focal parameter) versus 'a' and 'b' in ellipses/hyperbolas (axis lengths).
Geometric Understanding:
  • Focal Chord Direction: The latus rectum is always perpendicular to the major axis, not parallel. This perpendicular relationship is crucial for correct application.
  • Focus Location: Ensure proper focus identification, especially in ellipses and hyperbolas where multiple foci exist.
Calculation Accuracy:
  • Unit Consistency: Maintain consistent units throughout calculations, especially when working with real-world applications.
  • Sign Conventions: Pay attention to coordinate system orientations and maintain proper sign conventions for parameters.
Verification Methods:
  • Geometric Validation: Cross-check calculated values against known geometric properties and relationships.
  • Dimensional Analysis: Verify that results have correct dimensional units (length for latus rectum).

Error Prevention Examples

  • Incorrect: Using 2b²/a for parabolas (this applies only to ellipses/hyperbolas)
  • Correct: Using 4p for parabolas where p is the focal parameter
  • Mistake: Confusing semi-major with major axis (use a, not 2a in formulas)
  • Verification: For ellipse a=5, b=3: latus rectum = 2(9)/5 = 3.6, not 2(3)/5

Mathematical Derivation and Advanced Examples of Latus Rectum

  • Rigorous mathematical derivation from conic equations
  • Advanced applications in analytical geometry
  • Relationship to other conic section properties
The mathematical foundation of latus rectum calculations stems from the fundamental definitions of conic sections and their geometric properties in coordinate geometry.
Parabola Derivation:
Starting with the standard parabola y² = 4px, the focus is at (p, 0). The latus rectum is the vertical chord through this focus. Substituting x = p into the equation: y² = 4p(p) = 4p². Therefore, y = ±2p, giving a total chord length of 4p.
Ellipse Derivation:
For an ellipse x²/a² + y²/b² = 1, the foci are at (±c, 0) where c² = a² - b². The latus rectum passes through (c, 0) vertically. Substituting x = c: c²/a² + y²/b² = 1, solving for y: y² = b²(1 - c²/a²) = b²(b²/a²) = b⁴/a². Therefore, y = ±b²/a, giving total length 2b²/a.
Hyperbola Derivation:
For hyperbola x²/a² - y²/b² = 1, the foci are at (±c, 0) where c² = a² + b². Following similar substitution at x = c: c²/a² - y²/b² = 1, solving: y² = b²(c²/a² - 1) = b²(b²/a²) = b⁴/a². The result is identical to the ellipse: latus rectum = 2b²/a.
Eccentricity Relationships:
The latus rectum relates to eccentricity e through: For ellipse/hyperbola, latus rectum = a(1-e²) for ellipse and a(e²-1) for hyperbola, providing alternative calculation methods.
Parametric Representations:
In parametric form, the latus rectum endpoints can be expressed directly, providing computational advantages for complex geometric analysis and computer graphics applications.

Advanced Mathematical Examples

  • Parabola y² = 12x: p = 3, latus rectum = 4(3) = 12, endpoints at (3, ±6)
  • Ellipse x²/100 + y²/36 = 1: a = 10, b = 6, latus rectum = 2(36)/10 = 7.2
  • Hyperbola x²/25 - y²/144 = 1: a = 5, b = 12, latus rectum = 2(144)/5 = 57.6
  • Eccentricity check: For ellipse e = 0.8, a = 5: latus rectum = 5(1-0.64) = 1.8