Latus Rectum Calculator - Focal Chord for Any Conic

A latus rectum calculator finds the length of the focal chord of a conic section, the chord through one focus that lies parallel to the directrix. For a parabola, ellipse, or hyperbola in standard position, it returns the chord length, the focal distance, and the two endpoint coordinates in one pass.

• • Conic-section homework: Confirm the focal chord length given p or the semi-axes a and b.
• • Parabolic reflector design: Verify the width of a parabolic mirror at the focal plane from the focal length.
• • Orbital mechanics sanity checks: Use the semi-latus rectum p = a(1 - e^2) to cross-check a semi-major axis and eccentricity.
• • Graphing the focal chord: Plot the two endpoints of the latus rectum to add the focal chord to a sketch of the conic.

The page shows the focal distance, the chord length, and the two endpoint coordinates alongside the formula, so you can see exactly which arithmetic produced the result.

If you need the full vertex, focus, and directrix for a parabola in vertex form, the Parabola Calculator page handles the standard-form and vertex-form cases side by side.

The page picks one of three closed-form expressions depending on the conic you select, then reports the chord length, the focal distance, and the two endpoint coordinates.

• p: Focal length of a parabola, the distance from the vertex to the focus.
• a: Semi-major axis of an ellipse or the transverse semi-axis of a hyperbola.
• b: Semi-minor axis of an ellipse or the conjugate semi-axis of a hyperbola.
• c = sqrt(a^2 - b^2): Focal distance of an ellipse: positive when a is the larger of the two semi-axes.
• c = sqrt(a^2 + b^2): Focal distance of a hyperbola: always larger than a.
• L = 4p or 2b^2 / a: Latus rectum length, the chord through the focus parallel to the directrix.

For a parabola, the focal chord lies on the line x = p. For an ellipse or hyperbola, it lies on x = c, with half-length b^2 / a.

According to Wolfram MathWorld, the latus rectum of a conic is the chord through a focus parallel to the directrix, and for a parabola y^2 = 4px it has length 4p, for an ellipse x^2/a^2 + y^2/b^2 = 1 it has length 2b^2/a, and the same expression 2b^2/a holds for a hyperbola in standard position.

The semi-axes a and b that drive the latus rectum of an ellipse are the same a and b that drive its area, and the Ellipse Area page applies the matching pi a b formula to those inputs.

Four ideas explain why the focal chord has the length it does and what the result really means on the conic.

If you know the latus rectum length, the focal distance, and the endpoints, you can sketch the conic accurately without drawing the full curve. A hyperbola has one latus rectum on each branch, with the two chords translated by 2c along the x-axis.

The standard parabola form y^2 = 4px is a quadratic relation in y, so the same focal length p that sets the chord length also sets where the curve opens, and the Quadratic Formula Calculator page solves y^2 = 4px for x at any y in the same coordinate system this page uses.

Six short steps cover all three conics, from the default parabola to a hyperbola with a large conjugate axis.

1. 1 Pick the conic: Use the Conic dropdown to choose parabola, ellipse, or hyperbola. The page shows the right input fields and the right formula.
2. 2 Enter the focal length p (parabola): For a parabola, type the vertex-to-focus distance. The default of p = 1 reproduces y^2 = 4x with latus rectum 4.
3. 3 Enter a and b (ellipse or hyperbola): Type the semi-major axis a and the semi-minor (or conjugate) axis b. The default a = 5, b = 3 gives the standard 5-3 ellipse.
4. 4 Read the latus rectum length: The primary output is the chord length L, updated as you type. For a parabola it is 4p, for an ellipse or hyperbola it is 2b^2 / a.
5. 5 Check the focal distance and endpoints: Read the focal distance p or c and the two endpoint coordinates, which sit on x = p or x = c so a unit error is easy to spot.
6. 6 Reset or change the conic: Click Reset to return to the default parabola. Switching the dropdown keeps the other inputs, so you can compare conics in two clicks.

Try a hyperbola with a = 3 and b = 4. The page shows L = 2 * 16 / 3 = 10.6667, c = 5, and endpoints (5, 5.3333) and (5, -5.3333). The half-chord 5.3333 is exactly b^2 / a.

After reading the two endpoint coordinates, the focal chord is just a line segment between them, and the Length of a Line Segment Calculator page confirms the chord length, the midpoint, and the slope of the chord in one step from the same pair of points.

These benefits matter most when you are working a conic-section problem by hand and need a quick, trustworthy check.

• • Skip the arithmetic on focal chords: Manual focal-chord problems are easy to get wrong on the squaring step in 2b^2 / a. The calculator handles the squaring and the division.
• • See the formula for the active conic: The page shows the closed-form expression: 4p for a parabola and 2b^2 / a for an ellipse or hyperbola.
• • Get the focal distance and endpoints in one pass: The page returns the focal distance, the chord length, and the endpoint coordinates from the same inputs.
• • Handle the three conics from one place: Switching the dropdown is faster than opening three separate tools, and the output fields are the same in all three modes.
• • Spot unit or input errors quickly: Because the endpoints must lie on x = p or x = c, a wrong sign or wrong unit shows up immediately in the endpoint coordinates.

The page is most useful as a check, not as a replacement for understanding the geometry. Use it to confirm a homework answer, sanity-check an optics calculation, or pre-validate the parameters of a parabolic reflector before you hand them to a longer script.

When the ellipse is not centred at the origin, the focal chord shifts by the centre coordinates, and the Center of Ellipse Calculator page finds the centre from the standard-form equation so the latus rectum can be placed correctly.

The closed-form expression is the same in every case, but a few factors change how the result should be read.

• • The page covers the three conics in standard position only. A parabola with a vertex at (h, k) or an ellipse with a translated centre needs an extra translation step on the endpoint coordinates.
• • The hyperbola case assumes the transverse axis is horizontal. For a vertical transverse axis, swap the roles of a and b, and the endpoints then lie on the line y = c instead of x = c.
• • The page reports a single focal distance c for an ellipse or hyperbola. For the left focus, the sign on the endpoint x-coordinate must be flipped, but the chord length is unchanged.

According to Wikipedia, the latus rectum of a parabola in standard form y^2 = 4px has endpoints at (p, +/- 2p), and for an ellipse x^2/a^2 + y^2/b^2 = 1 each latus rectum has endpoints at (+/- ae, +/- b^2/a) where e is the eccentricity.

The endpoints of the latus rectum are a pair of points, so the 2D Distance Calculator page is the easiest way to confirm the chord length from the endpoint coordinates without re-deriving the focal distance.