Parabola

Vertex at origin, axis along positive x-axis. Focus at (a, 0), directrix x = −a, latus rectum x = a.

Vertex at origin, axis along negative x-axis. Focus at (−a, 0), directrix x = a.

Vertex at origin, axis along positive y-axis. Focus at (0, a), directrix y = −a.

Vertex at origin, axis along negative y-axis. Focus at (0, −a), directrix y = a.

Vertex at (h, k), axis parallel to x-axis. All properties of y² = 4ax apply with origin shifted to (h, k). Similarly (x−h)² = 4a(y−k) for axis parallel to y-axis.

Distance from any point P(x₁, y₁) on y² = 4ax to the focus S(a, 0). Equals the distance from P to the directrix — the defining property of a parabola.

The latus rectum is the chord through the focus perpendicular to the axis. Its endpoints are (a, 2a) and (a, −2a). Half the latus rectum (semi-latus rectum) = 2a.

Every point on y² = 4ax can be written as (at², 2at) for a unique t. Parametric form simplifies tangent, normal, and chord problems considerably.

If (at₁², 2at₁) and (at₂², 2at₂) are the endpoints of a focal chord of y² = 4ax, then t₁t₂ = −1. Equivalently, one parameter is the negative reciprocal of the other.

Length of the focal chord with parameter t at one end (and −1/t at the other). Minimum value 4a (the latus rectum) is achieved when t = ±1.

The tangents to y² = 4ax at parameters t₁ and t₂ intersect at (at₁t₂, a(t₁+t₂)). For a focal chord (t₁t₂ = −1), the tangents meet on the directrix.

Tangent to y² = 4ax at the point (x₁, y₁) lying on it. Obtained by the T = 0 rule: replace y² → yy₁, x → (x+x₁)/2.

Tangent to y² = 4ax at the point (at², 2at). Slope of this tangent is 1/t.

Tangent to y² = 4ax with slope m ≠ 0. One tangent exists for each slope (unlike a circle, which has two). Point of contact is (a/m², 2a/m).

Line y = mx + c is tangent to y² = 4ax if and only if c = a/m. Unlike circles, only one value of c works for each slope m.

For y² = 4ax, the foot of the perpendicular drawn from the focus to any tangent always lies on the y-axis (the tangent at the vertex). Equivalently, the tangent at the vertex is the locus of feet of perpendiculars from the focus.

Normal to y² = 4ax at (x₁, y₁). Slope of normal = −y₁/2a (negative reciprocal of the tangent slope 2a/y₁).

Normal to y² = 4ax at (at², 2at). Slope of this normal is −t. The foot of the normal on the axis is at (2a + at², 0).

Normal to y² = 4ax with slope m. Up to three normals can be drawn from a given external point, corresponding to three real roots of the cubic in m.

Chord joining the two points of tangency when tangents are drawn from external point (x₁, y₁). Same algebraic form as the tangent at a point — context determines which.

Combined equation of the two tangents from external point (x₁, y₁) to y² = 4ax, where S = y²−4ax, S₁ = y₁²−4ax₁, T = yy₁−2a(x+x₁).

Equation of the chord of y² = 4ax whose midpoint is (x₁, y₁). Explicitly: yy₁ − 2a(x+x₁) = y₁² − 4ax₁.

Polar of point (x₁, y₁) with respect to y² = 4ax. If (x₁, y₁) is on the parabola, the polar is the tangent there. If external, it is the chord of contact. La Hire's theorem holds.

If three normals from an external point meet y² = 4ax at parameters t₁, t₂, t₃, then t₁+t₂+t₃ = 0. Equivalently, the sum of the slopes of the three normals is zero.

Three normals can be drawn from (h, k) to y² = 4ax with slopes m₁, m₂, m₃ satisfying this cubic. By Vieta's: m₁+m₂+m₃ = 0, m₁m₂+m₂m₃+m₃m₁ = (2a−h)/a, m₁m₂m₃ = −k/a.