Latus rectum of a parabola

The latus rectum of a parabola is the chord through the focus perpendicular to the axis, with both ends on the parabola.

For y² = 4ax, each end of the latus rectum is as far from the focus as from the directrix, which is 2a away. So each half has length 2a, and the latus rectum has length 4a; its ends are (a, 2a) and (a, −2a).

y² = 8x: the y² term puts the axis along the x-axis, the positive coefficient of x opens it to the right, and 4a = 8 gives a = 2. Focus (2, 0), directrix x = −2, latus rectum 8.

Going the other way: focus (2, 0) and directrix x = −2 give y² = 8x; vertex (0, 0) and focus (0, 2) give x² = 8y.

A parabola with vertex at the origin, symmetric about the y-axis and passing through (2, −3), must open downwards, so x² = −4ay. Then 4 = 12a, a = 1/3, and the equation is 3x² = −4y.

For x² = −12y: axis the y-axis, opening down, a = 3, focus (0, −3), directrix y = 3, latus rectum 12.