the latus rectum
/ LAY-tus REK-tum /
If you stand at the focus of a conic and look straight across — along a line parallel to the directrix, perpendicular to the axis — the chord of the curve you span is the latus rectum (Latin for 'straight side'). It is a tidy, fixed measurement of how 'wide' the conic is right at the focus, and it appears constantly in the conics' formulas.
Precisely, the latus rectum is the chord that passes through a focus and is perpendicular to the major (or transverse) axis, with its endpoints on the curve. Its length has clean formulas: for a parabola x^2 = 4py the latus rectum has length |4p|; for an ellipse or hyperbola with semi-axes a and b it has length 2b^2/a. Half of it, the semi-latus rectum, is written l = b^2/a (for the parabola l = 2p), and this single number l is the natural 'size scale' of the conic — it is exactly the value that appears in the polar equation r = l / (1 +- e cos theta).
Why bother with it? Because the semi-latus rectum together with the eccentricity e completely determines a conic's shape and size in the cleanest way. In orbital mechanics l and e are the two numbers an astronomer reads off to describe an orbit, and the latus rectum gives you two more exact points (the ends of the chord) to plot when sketching, located right beside the focus.
For the parabola x^2 = 8y, here 4p = 8, so the latus rectum has length 8; its endpoints are (-4, 2) and (4, 2), level with the focus (0, 2). For the ellipse x^2/25 + y^2/9 = 1, a = 5 and b^2 = 9, so the latus rectum length is 2b^2/a = 18/5 = 3.6 and the semi-latus rectum is l = 9/5 = 1.8.
The latus rectum is the focal chord across the axis; its half l = b^2/a scales the polar form.
'Latus rectum' (plural latera recta) names the full chord; the value b^2/a is the semi-latus rectum, exactly half. Mixing them up doubles your answer, so check whether a formula wants the chord or its half.