Latus Rectum Properties
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Direct answer
The latus rectum (LR) is the focal chord perpendicular to the axis — for a parabola or ellipse, the shortest chord drawn through the focus — and its length is the first number examiners ask for. Parabola y² = 4ax: length 4a with endpoints (a, ±2a). Ellipse x²/a² + y²/b² = 1 (a > b): length 2b²/a = 2a(1 − e²) with endpoints (±ae, ±b²/a). Hyperbola x²/a² − y²/b² = 1: length 2b²/a = 2a(e² − 1), same endpoint pattern. Each formula encodes eccentricity, and in the polar description r = p/(1 + e cosθ) the semi-latus rectum is precisely the p of the conic.
What you must remember
- Parabola: LR = 4a, endpoints (a, ±2a); the tangents at these ends meet at (−a, 0), on the directrix; among focal chords the LR is the shortest.
- Ellipse: LR = 2b²/a = 2a(1 − e²); two latus rectums at x = ±ae, endpoints (±ae, ±b²/a).
- Hyperbola: LR = 2b²/a = 2a(e² − 1), endpoints (±ae, ±b²/a); the conjugate hyperbola's LR is 2a²/b.
- Polar unification: with a focus as pole, every conic reads r = p/(1 + e cosθ) with semi-latus rectum p: 2a for the parabola, b²/a for ellipse and hyperbola.
- Eccentricity retrievals: b² = a²(1 − e²) for the ellipse and b² = a²(e² − 1) for the hyperbola — the sign before e² is the whole difference.
- Focal chord comparison: a parabola's focal chord through parameter t has length a(t + 1/t)² ≥ 4a, equality at t = ±1 — the LR again.
- Defining distance: the focal distance of an LR endpoint equals the semi-latus rectum itself, which is why p controls the conic's opening at the focus.
Numbers from one ellipse
Take x²/25 + y²/16 = 1. Here a = 5, b = 4, so c = √(25 − 16) = 3 and e = 3/5. The LR length is 2b²/a = 32/5, and the endpoints are (±ae, ±b²/a) = (±3, ±16/5). Run the cross-checks an examiner rewards: 2a(1 − e²) = 10(1 − 9/25) = 10 × 16/25 = 32/5 ✓, and ae = 3 places the latus rectums directly above and below the foci (±3, 0) ✓.
The parabola twin: y² = 12x has 4a = 12, so a = 3, LR = 12, endpoints (3, ±6). The tangents there correspond to t = ±1: ty = x + 3t² gives y = x + 3 and −y = x + 3, meeting at (−3, 0) — on the directrix x = −3, exactly as the theory promises. Two curves, one page, and every formula in the bullet list exercised once with arithmetic small enough to verify mentally.
Length mix-ups
Main asks LR lengths or endpoint coordinates directly; Advanced hides the LR inside eccentricity or focal-chord questions — for instance, "a hyperbola and its conjugate share a and b; compare their latus rectums" (they differ: 2b²/a versus 2a²/b, a favourite true/false). The traps: writing 2a(1 − e²) for a hyperbola — the sign flip is the error; quoting 4a for a non-parabola; giving the conjugate hyperbola the original's LR; and placing the ellipse's LR endpoints at x = ±c without noting that for the ellipse c = ae, so the discipline of writing ±ae is what transfers to shifted or rotated settings. A final slip worth naming: the LR is perpendicular to the major or transverse axis — rotating the axes rotates the chord, and questions that swap the axis roles invalidate memorised coordinates.
Frequently asked questions
What is the latus rectum of y² = 4ax?
The chord x = a through the focus: length 4a with endpoints (a, ±2a); the tangents at its ends meet on the directrix.
How do the ellipse and hyperbola LR formulas differ?
Both read 2b²/a, but b² = a²(1 − e²) for the ellipse and a²(e² − 1) for the hyperbola — the sign in front of e² is the difference.
Are the latus rectums of a hyperbola and its conjugate equal?
No — the conjugate swaps a and b, so its LR is 2a²/b.
What does the semi-latus rectum mean in polar form?
It is the p in r = p/(1 + e cosθ): the focal distance measured perpendicular to the axis of the conic.
Which focal chord of a parabola is shortest?
The latus rectum itself: a focal chord through parameter t has length a(t + 1/t)² ≥ 4a, with equality at t = ±1.