Latus Rectum Calculator
Free Latus rectum Calculator for coordinate geometry. Enter values to get step-by-step solutions with formulas and graphs.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Latus Rectum Calculator
Calculator
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Formula: Parabola: L = 4a | Ellipse: L = 2b²/a | Hyperbola: L = 2b²/a
Worked example — Latus Rectum: 16 | Focus: (4, 0) | Directrix: x = -4
Formula
Parabola: L = 4a | Ellipse: L = 2b²/a | Hyperbola: L = 2b²/a
The latus rectum L is the length of the chord through the focus perpendicular to the principal axis. For a parabola y² = 4ax, L = 4a. For an ellipse or hyperbola, L = 2b²/a where a and b are the semi-axes.
Worked Examples
Example 1: Parabola Latus Rectum
Problem:Find the latus rectum of the parabola y² = 16x.
Solution:Compare with y² = 4ax: 4a = 16, so a = 4 Latus rectum = 4a = 16 Semi-latus rectum = 2a = 8 Focus at (4, 0) Directrix: x = -4 Endpoints 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 x²/25 + y²/9 = 1.
Solution:a = 5 (semi-major), b = 3 (semi-minor) Latus rectum = 2b²/a = 2(9)/5 = 18/5 = 3.6 Semi-latus rectum = 9/5 = 1.8 c = sqrt(25 - 9) = sqrt(16) = 4 Eccentricity = c/a = 4/5 = 0.8 Foci at (4, 0) and (-4, 0)
Result:Latus Rectum: 3.6 | Eccentricity: 0.8 | Foci: (±4, 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 y² = 4ax, the latus rectum length is 4a. For an ellipse x²/a² + y²/b² = 1, it equals 2b²/a. For a hyperbola x²/a² - y²/b² = 1, it also equals 2b²/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 y² = 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 x² = 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, y² = 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 x²/a² + y²/b² = 1, where a is the semi-major axis and b is the semi-minor axis, the latus rectum length is 2b²/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 b²/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 = b²/a. For a parabola, l = 2a. For a hyperbola, l = b²/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 - e²), 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.
What is the latus rectum of a hyperbola?
For a hyperbola x²/a² - y²/b² = 1, the latus rectum length is 2b²/a, the same formula as for an ellipse. Here a is the semi-transverse axis (half the distance between vertices) and b is the semi-conjugate axis. The latus rectum endpoints lie on the hyperbola at x = c (the focus location), where c = sqrt(a² + b²). The latus rectum is always shorter than the transverse axis (2a). Unlike an ellipse where the latus rectum is bounded by the shape, the hyperbola's latus rectum chord connects two points on the same branch of the hyperbola. The semi-latus rectum appears in the polar equation of the hyperbola relative to either focus.
How do conic sections appear in real-world applications?
Conic sections and their latus rectum parameters appear throughout science and engineering. Planetary orbits are ellipses with the Sun at one focus, and the semi-latus rectum determines the orbital size. Parabolic mirrors focus incoming parallel light to a single point (the focus), with the latus rectum determining the mirror width at the focal plane. Satellite dishes and car headlight reflectors use parabolic shapes for this property. Hyperbolic navigation systems (like the historical LORAN) use the properties of hyperbolas for positioning. Elliptical rooms create whispering galleries where sound from one focus reflects perfectly to the other focus. In each case, the latus rectum helps engineers calculate dimensions and performance.
Can the latus rectum be used to find the equation of a conic?
Yes, knowing the latus rectum along with the type and eccentricity (or focus position) is sufficient to determine the complete equation of a conic section. Given the semi-latus rectum l and eccentricity e, you can write the polar equation r = l / (1 + e * cos(theta)) immediately. For a parabola (e = 1) with latus rectum L, the parameter a = L/4, giving y² = Lx. For an ellipse with semi-latus rectum l and eccentricity e, find a = l / (1 - e²) and b = a * sqrt(1 - e²). This approach is commonly used in astrodynamics, where orbital elements are given as semi-latus rectum and eccentricity rather than semi-major and semi-minor axes.
What is the difference between the latus rectum and the directrix?
The latus rectum and the directrix are both important features of conic sections, but they serve different roles. The directrix is a line, while the latus rectum is a line segment (chord). The directrix defines the conic through the focus-directrix property: the ratio of the distance from any point to the focus versus the distance to the directrix equals the eccentricity. The latus rectum passes through the focus and is perpendicular to the axis connecting the focus to the directrix. For a parabola y² = 4ax, the directrix is the line x = -a, while the latus rectum is the chord at x = a with length 4a. Together, the directrix and latus rectum provide complementary geometric information about the conic's shape and orientation.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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