Latus Rectum Calculator
Free Latus rectum Calculator for coordinate geometry. Enter values to get step-by-step solutions with formulas and graphs.
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.