Latus Rectum

High School

Definition

It is a segment of a line that perpendicular to the principal axis of a parabola, hyperbola, or ellipse. It starts and ends on the curve, so its endpoints sit on the curve. Different conics have different ways of determining its latus rectum.

Worked examples

\(y^2 = 4px\) parabola: latus rectum length \(= 4p\)

For a parabola opening right with vertex at origin, the latus rectum passes through the focus at \(x = p\).

\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) ellipse: latus rectum length \(= \frac{2b^2}{a}\)

The latus rectum is perpendicular to the major axis and passes through each focus.

\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) hyperbola: latus rectum length \(= \frac{2b^2}{a}\)

Each branch of the hyperbola has a latus rectum through its focus, perpendicular to the transverse axis.

Common mistakes

The latus rectum is parallel to the principal axis → The latus rectum is perpendicular to the principal axis By definition, the latus rectum crosses the axis at right angles, not runs alongside it.
Using \(4p\) for the latus rectum of an ellipse → Use \(\frac{2b^2}{a}\) for an ellipse Each conic has its own formula; \(4p\) is specific to parabolas only.
Placing the latus rectum at the vertex instead of the focus → The latus rectum passes through the focus The segment must go through the focus and be perpendicular to the axis.

Where you'll use it next

You'll use the latus rectum when sketching and analyzing conics in analytic geometry, finding focal widths for telescope and satellite dish designs, and solving optimization problems involving parabolic and elliptical paths in calculus and physics.

Found in 1 StudyPug lesson

Conics - Parabola

Grade 12 Math

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