Latus Rectum

Latus Rectum

The line segment through a focus of a conic section, perpendicular to the major axis, which has both endpoints on the curve.

Note: The length of a parabola's latus rectum is 4p, where p is the distance from the focus to the vertex.

Upward-opening parabola with labeled focus point and horizontal latus rectum line segment passing through the focus.

Key Formula

\text{Parabola: } L = 4p \qquad \text{Ellipse: } L = \frac{2b^2}{a} \qquad \text{Hyperbola: } L = \frac{2b^2}{a}

Where:

$L$ = Length of the latus rectum
$p$ = Distance from the vertex to the focus of a parabola
$a$ = Semi-major axis length (ellipse) or semi-transverse axis length (hyperbola)
$b$ = Semi-minor axis length (ellipse) or semi-conjugate axis length (hyperbola)

Worked Example

Problem: Find the length of the latus rectum of the parabola y² = 12x.

Step 1: Write the parabola in standard form y² = 4px and identify 4p.

y^2 = 4px \implies 4p = 12

Step 2: Solve for p.

p = \frac{12}{4} = 3

Step 3: The length of the latus rectum for a parabola is 4p.

L = 4p = 4(3) = 12

Step 4: Verify by finding the endpoints. The focus is at (3, 0). Substitute x = 3 into y² = 12x.

y^2 = 12(3) = 36 \implies y = \pm 6

Step 5: The endpoints are (3, 6) and (3, −6). The distance between them is 6 − (−6) = 12, which confirms our answer.

L = 6 - (-6) = 12

Answer: The length of the latus rectum is 12 units.

Another Example

This example applies the latus rectum formula to an ellipse rather than a parabola, showing how the formula differs between conic sections.

Problem: Find the length of the latus rectum of the ellipse x²/25 + y²/9 = 1.

Step 1: Identify a² and b² from the standard form x²/a² + y²/b² = 1, where a > b.

a^2 = 25,\quad b^2 = 9 \implies a = 5,\quad b = 3

Step 2: Apply the latus rectum formula for an ellipse.

L = \frac{2b^2}{a} = \frac{2(9)}{5} = \frac{18}{5}

Step 3: Compute the decimal value for clarity.

L = \frac{18}{5} = 3.6

Answer: The length of the latus rectum is 18/5 = 3.6 units.

Frequently Asked Questions

What is the difference between the latus rectum of a parabola and an ellipse?

For a parabola, the latus rectum length is 4p, where p is the vertex-to-focus distance. For an ellipse, the length is 2b²/a, where a is the semi-major axis and b is the semi-minor axis. Both line segments pass through a focus perpendicular to the main axis, but the formulas reflect the different geometry of each curve.

Why is the latus rectum important?

The latus rectum gives a direct measure of how "wide" a conic section is at its focus. For a parabola, a larger latus rectum means a wider, more open curve. In optics and engineering, this measurement determines properties like the spread of a satellite dish or the curvature of a reflective mirror.

How do you find the endpoints of the latus rectum?

First locate the focus of the conic. Then draw a vertical (or horizontal) line through the focus, perpendicular to the axis of symmetry. Substitute the focus coordinate into the equation of the conic and solve for the other variable. The two solutions give the endpoints of the latus rectum.

Latus Rectum (Parabola) vs. Latus Rectum (Ellipse)

	Latus Rectum (Parabola)	Latus Rectum (Ellipse)
Definition	Chord through the focus, perpendicular to the axis of symmetry	Chord through a focus, perpendicular to the major axis
Formula	L = 4p	L = 2b²/a
Number of latus recta	One (single focus)	Two (one per focus), each with the same length
What it reveals	How wide/narrow the parabola opens	The width of the ellipse at each focus

Why It Matters

You encounter the latus rectum in precalculus and calculus courses whenever you study conic sections. It appears in problems about parabolic reflectors, satellite dishes, and telescope mirrors, where the width at the focus determines the device's performance. Understanding it also helps you quickly sketch accurate graphs of parabolas, ellipses, and hyperbolas by knowing the curve's width at the focus.

Common Mistakes

Mistake: Using 2p instead of 4p for the latus rectum of a parabola.

Correction: The semi-latus rectum (from the focus to one endpoint) is 2p. The full latus rectum spans both endpoints and equals 4p. Always check whether a problem asks for the full length or the semi-latus rectum.

Mistake: Confusing a and b when computing the latus rectum of an ellipse.

Correction: In the formula L = 2b²/a, the value a must be the semi-major axis (the larger one) and b the semi-minor axis. If you swap them, you get the wrong answer. Always confirm which denominator is larger in x²/a² + y²/b² = 1.

Related Terms

Parabola — Conic where latus rectum equals 4p
Focus of a Parabola — Point the latus rectum passes through
Vertex of a Parabola — Defines distance p to the focus
Conic Sections — Family of curves with a latus rectum
Foci of an Ellipse — Each focus has its own latus rectum
Major Axis of an Ellipse — Latus rectum is perpendicular to this axis
Perpendicular — Orientation of latus rectum relative to axis
Line Segment — The latus rectum is a specific chord