In mathematics, the latus rectum of a conic section is the chord parallel to the directrix and passing through the single focus, or one of the two foci. By extension, the length of the latus rectum is also referred to as the latus rectum and written <math>\scriptstyle 2\ell\,</math>.
Semi-latus rectum and latus rectum in the case of an ellipse
The semi-latus rectum is half the latus rectum. By extension the length of the semi-latus rectum is also referred to as the semi-latus rectum and written <math>\scriptstyle\ell\,</math>.
| conic section | equation | semi-latus rectum <math>\left (\scriptstyle\ell \right )</math> | latus rectum <math>\left ( \scriptstyle 2\ell \right )\,</math> |
|---|---|---|---|
| circle | <math>x^2+y^2=r^2 \,</math> | <math> r \,</math> | <math> 2r \,</math> |
| ellipse | <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math> | <math>\frac{b^2}{a}</math> | <math>\frac{2b^2}{a}</math> |
| parabola | <math>y^2=4ax \,</math> | <math>2a \,</math> | <math>4a \,</math> |
| hyperbola | <math>\frac{x^2}{a^2}-\frac{y^2}{b^2}=1</math> | <math>\frac{b^2}{a}</math> | <math>\frac{2b^2}{a}</math> |
The semi-latus rectum appears in the equation of a conic section in polar coordinates:
- <math>r=\frac{\ell}{1+\varepsilon \cos(\phi)}</math>
