Latus Rectum Calculator
Calculate the latus rectum length and endpoints for parabolas, hyperbolas, and ellipses
Calculate Latus Rectum
Parabola Parameters
Standard Form: y = Ax² + Bx + C
Or Direct Parameter
Use this if you know the parameter 'a' directly
Latus Rectum Results
Formula used: lr = 4a (where a is the distance from vertex to focus)
Analysis
Example Calculations
Parabola Example
Equation: y = 4x² - 2x + 6
Parameters: A = 4, B = -2, C = 6
Vertex: (0.25, 5.75)
Focus distance (a): 1/(4×4) = 0.0625
Latus rectum: 4 × 0.0625 = 0.25
Hyperbola Example
Equation: (x-2)²/9 - y²/4 = 1
Parameters: a = 3, b = 2
Latus rectum: 2 × 2² / 3 = 8/3 ≈ 2.667
Ellipse Example
Equation: x²/25 + y²/7 = 1
Parameters: a = 5, b = √7 ≈ 2.646
Latus rectum: 2 × 7 / 5 = 2.8
Conic Sections
Parabola
lr = 4a
One focus, one latus rectum
Hyperbola
lr = 2b²/a
Two foci, two latus recta
Ellipse
lr = 2b²/a
Two foci, two latus recta
Key Properties
Latus rectum passes through the focus
It's perpendicular to the major axis
Endpoints lie on the conic curve
Latin for "straight side"
Understanding the Latus Rectum
What is the Latus Rectum?
The latus rectum is a line segment that passes through a focus of a conic section and is perpendicular to the major axis. The endpoints of this segment lie on the curve of the conic section.
Etymology
The term comes from Latin: "latus" meaning "side" and "rectum" meaning "straight". It was first used by mathematician Johannes Kepler in the early 17th century.
Applications
- •Antenna design and satellite dishes
- •Orbital mechanics and astronomy
- •Optical systems and lens design
- •Bridge and arch construction
Formula Explanations
Parabola
lr = 4a
Where 'a' is the distance from the vertex to the focus. For a parabola y = Ax² + Bx + C, we have a = 1/(4A).
Hyperbola & Ellipse
lr = 2b²/a
Where 'a' is the semi-major axis and 'b' is the semi-minor axis. Both shapes use the same formula but have different geometric interpretations.
Linear Eccentricity
- Hyperbola: c = √(a² + b²)
- Ellipse: c = √(|a² - b²|)
- Used to locate the foci and endpoints