Latus Rectum Calculator
Free Latus rectum Calculator for coordinate geometry. Enter values to get step-by-step solutions with formulas and graphs.
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The latus rectum L is the length of the chord through the focus perpendicular to the principal axis. For a parabola y\u00B2 = 4ax, L = 4a. For an ellipse or hyperbola, L = 2b\u00B2/a where a and b are the semi-axes.
Last reviewed: December 2025
Worked Examples
Example 1: Parabola Latus Rectum
Example 2: Ellipse Latus Rectum
Background & Theory
The Latus Rectum Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Latus Rectum Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Sources & References
Formula
Parabola: L = 4a | Ellipse: L = 2b\u00B2/a | Hyperbola: L = 2b\u00B2/a
The latus rectum L is the length of the chord through the focus perpendicular to the principal axis. For a parabola y\u00B2 = 4ax, L = 4a. For an ellipse or hyperbola, L = 2b\u00B2/a where a and b are the semi-axes.
Worked Examples
Example 1: Parabola Latus Rectum
Problem: Find the latus rectum of the parabola y\u00B2 = 16x.
Solution: Compare with y\u00B2 = 4ax: 4a = 16, so a = 4\nLatus rectum = 4a = 16\nSemi-latus rectum = 2a = 8\nFocus at (4, 0)\nDirectrix: x = -4\nEndpoints of latus rectum: (4, 8) and (4, -8)
Result: Latus Rectum: 16 | Focus: (4, 0) | Directrix: x = -4
Example 2: Ellipse Latus Rectum
Problem: Find the latus rectum of the ellipse x\u00B2/25 + y\u00B2/9 = 1.
Solution: a = 5 (semi-major), b = 3 (semi-minor)\nLatus rectum = 2b\u00B2/a = 2(9)/5 = 18/5 = 3.6\nSemi-latus rectum = 9/5 = 1.8\nc = sqrt(25 - 9) = sqrt(16) = 4\nEccentricity = c/a = 4/5 = 0.8\nFoci at (4, 0) and (-4, 0)
Result: Latus Rectum: 3.6 | Eccentricity: 0.8 | Foci: (\u00B14, 0)
Frequently Asked Questions
What is the latus rectum of a conic section?
The latus rectum is a chord of a conic section that passes through the focus and is perpendicular to the major axis (or the axis of symmetry in the case of a parabola). Its length provides important information about the shape and curvature of the conic near the focus. For a parabola y\u00B2 = 4ax, the latus rectum length is 4a. For an ellipse x\u00B2/a\u00B2 + y\u00B2/b\u00B2 = 1, it equals 2b\u00B2/a. For a hyperbola x\u00B2/a\u00B2 - y\u00B2/b\u00B2 = 1, it also equals 2b\u00B2/a. The semi-latus rectum (half the latus rectum) is frequently used in orbital mechanics to describe planetary orbits.
Why is the latus rectum important in conic sections?
The latus rectum is a fundamental parameter that characterizes the local curvature of a conic section near its focus. In orbital mechanics, the semi-latus rectum determines the shape of an orbit and appears directly in the polar equation of a conic: r = l / (1 + e * cos(theta)), where l is the semi-latus rectum and e is the eccentricity. The latus rectum also connects the geometric properties of the conic to its focal properties, making it essential for calculations involving reflectors, satellite orbits, and optical systems. A larger latus rectum indicates a wider, more open curve near the focus.
How do you find the latus rectum of a parabola?
For a parabola in the form y\u00B2 = 4ax, the latus rectum has length 4a, where a is the distance from the vertex to the focus. The endpoints of the latus rectum are at (a, 2a) and (a, -2a). If the parabola opens upward as x\u00B2 = 4ay, the latus rectum is horizontal with the same length 4a. To find the latus rectum from a general parabolic equation, first convert to standard form to identify the value of a. For example, y\u00B2 = 12x gives 4a = 12, so a = 3 and the latus rectum length is 12. The latus rectum of a parabola is the only chord through the focus perpendicular to the axis.
How is the latus rectum of an ellipse calculated?
For an ellipse with equation x\u00B2/a\u00B2 + y\u00B2/b\u00B2 = 1, where a is the semi-major axis and b is the semi-minor axis, the latus rectum length is 2b\u00B2/a. This formula gives the total length of the chord through the focus perpendicular to the major axis. Since an ellipse has two foci, there are two latus rectum chords, both with the same length. As the eccentricity approaches zero (circle), the latus rectum approaches 2a = 2b, which is the diameter. As the eccentricity increases toward 1, the latus rectum decreases. The semi-latus rectum b\u00B2/a is used in the polar form of the ellipse equation and in Kepler's laws of planetary motion.
What is the semi-latus rectum and where is it used?
The semi-latus rectum is half the latus rectum and is commonly denoted by the letter l or p. For an ellipse, l = b\u00B2/a. For a parabola, l = 2a. For a hyperbola, l = b\u00B2/a. The semi-latus rectum appears prominently in the unified polar equation of all conics: r = l / (1 + e * cos(theta)), where e is the eccentricity. This makes it especially important in celestial mechanics, where planetary orbits are described as conic sections with the Sun at one focus. In Kepler's second law, the areal velocity of a planet is directly proportional to the square root of the semi-latus rectum. Engineers also use it in antenna and reflector design.
How does eccentricity relate to the latus rectum?
Eccentricity and the latus rectum together completely define a conic section. The eccentricity e determines the type of conic (e = 0 for circle, 0 < e < 1 for ellipse, e = 1 for parabola, e > 1 for hyperbola), while the latus rectum determines the scale. The polar equation r = l / (1 + e * cos(theta)) shows that the semi-latus rectum l is the distance from the focus to the conic when theta = 90 degrees. For an ellipse, l = a(1 - e\u00B2), connecting the semi-major axis and eccentricity to the semi-latus rectum. A higher eccentricity with the same semi-latus rectum produces a more elongated conic, while a larger semi-latus rectum with the same eccentricity produces a larger conic of the same shape.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy