Latus Rectum
Calculate the latus rectum length for ellipses, hyperbolas, and parabolas. Free online conic section calculator with formulas and step-by-step results for geometry students.
About This Calculator
A latus rectum calculator computes the length of the latus rectum -- a line segment through a focus of a conic section (ellipse, parabola, or hyperbola) that runs parallel to the directrix with endpoints on the curve. This mathematical tool is essential for students studying conic sections in analytic geometry, engineers working with reflective properties of parabolic antennas and ellipsoidal reflectors, and anyone analyzing the geometric properties of conic curves.
Latus Rectum Formulas
The latus rectum length depends on the conic section type. For an ellipse in standard form x^2/a^2 + y^2/b^2 = 1, the latus rectum is LR = 2b^2/a, where a is the semi-major axis and b is the semi-minor axis. Each ellipse has two latera recta -- one passing through each focus. For a hyperbola in standard form x^2/a^2 - y^2/b^2 = 1, the latus rectum uses the same formula LR = 2b^2/a, with a as the transverse axis and b as the conjugate axis. For a parabola in standard form y^2 = 4ax, the latus rectum formula simplifies to LR = 4a, where a is the focal length (distance from the vertex to the focus).
How to Use This Calculator
Start by selecting the conic section type from the dropdown -- ellipse, hyperbola, or parabola. For ellipse and hyperbola, enter the semi-major axis a and semi-minor axis b values. For parabola, enter the focal length a only (the b input is not needed since a parabola has no semi-minor axis). Click "Calculate Latus Rectum" to instantly get the latus rectum length along with the formula used for computation. The result updates in real time and supports shareable URL links -- simply copy the URL after calculating to share your results.
Who Uses the Latus Rectum Calculator?
This calculator serves high school and college students studying conic sections in algebra and precalculus courses, mathematics educators preparing lesson materials on analytic geometry, engineers designing parabolic reflectors and antennas, and physicists studying orbital mechanics where ellipses describe planetary orbits. The latus rectum is particularly important in the study of focal properties of reflective surfaces.
Frequently Asked Questions
What is the latus rectum of a conic section?
The latus rectum is a line segment that passes through a focus of a conic section and is parallel to the directrix. Its endpoints lie on the conic curve itself. The term comes from Latin, where 'latus' means 'side' and 'rectum' means 'straight'.
How do you find the latus rectum of an ellipse?
For an ellipse in standard form x^2/a^2 + y^2/b^2 = 1, the latus rectum length is LR = 2b^2/a, where a is the semi-major axis and b is the semi-minor axis. An ellipse has two latera recta, one through each focus.
What is the latus rectum formula for a parabola?
For a parabola in standard form y^2 = 4ax, the latus rectum length is LR = 4a, where a is the focal length (distance from the vertex to the focus).
How do you calculate the latus rectum of a hyperbola?
For a hyperbola in standard form x^2/a^2 - y^2/b^2 = 1, the latus rectum length is LR = 2b^2/a, where a is the transverse axis and b is the conjugate axis. A hyperbola has two latera recta, one through each focus.
Does a circle have a latus rectum?
Yes, a circle is a special case of an ellipse where a = b = r. Its latus rectum length is LR = 2r^2/r = 2r, which equals the diameter of the circle. However, circles are typically not discussed in terms of latus rectum since their foci coincide at the center.
What does 'latus rectum' mean in Latin?
Latus rectum is a Latin term where 'latus' means 'side' and 'rectum' means 'straight'. Together it translates to 'straight side', referring to the straight line segment through a focus of a conic section.
What are the endpoints of the latus rectum?
The endpoints of the latus rectum lie on the conic curve. For a horizontal ellipse centered at (h, k) with linear eccentricity c, the endpoints are at (h+/-c, k+/-LR/2). For a vertical parabola opening upward, endpoints are at (h+/-LR, k+LR/2).