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.

States that the distance of any point on the parabola from the focus equals its perpendicular distance from the directrix.

Gives the coordinates of the point of contact where the tangent of slope m touches y² = 4ax.

Gives the equation of the tangent with slope m to the vertical parabola x² = 4ay.