Parabola

A parabola is the locus of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The line through the focus perpendicular to the directrix is the axis of symmetry; the parabola crosses it at the vertex, halfway between focus and directrix. The four standard orientations place the vertex at the origin and align the axis with a coordinate axis, differing only in direction. The parameter a is the focus-to-vertex distance and controls how tightly the parabola opens.

Type of Parabola

The four standard parabolas all have their vertex at the origin; they differ in which axis they open along and in which direction the focus lies. The latus rectum is the focal chord perpendicular to the axis, with length 4a in every case. The directrix lies on the side of the vertex opposite the focus, at distance a.

Parabola	y2=4ax	y2=-4ax	x2=4ay	x2=-4ay
Vertex	(0,0)	(0,0)	(0,0)	(0,0)
Focus	(a,0)	(-a,0)	(0,a)	(0,-a)
Axis	y=0	y=0	x=0	x=0
Dirctix	x=-a	x=a	y=-a	y=a
Length of Latus Recutum (LR)	4a	4a	4a	4a
End Point of LR	(a,2a)&(a,-2a)	(-a,2a)&(-a,-2a)	(2a,a)&(-2a,a)	(2a,-a)&(-2a,-a)

Tangent To Parabola y²=4ax

A parabola has two tangent forms: one from a point of contact, one from a slope. The point form applies the T = 0 substitution y² → y y₁ and x → (x + x₁)/2, multiplied out. The contact point (x₁, y₁) must lie on the parabola, i.e. satisfy y₁² = 4a x₁.

The slope form gives the tangent of any non-zero slope m. Substituting y = mx + c into the parabola equation and demanding a double root yields c = a/m. Each non-zero slope has exactly one tangent; the slope-zero limit is the x-axis, which is tangent at the vertex.

Normal To Parabola y²=4ax

The normal at a contact point is perpendicular to the tangent there and passes through the point. Its slope is the negative reciprocal of the tangent slope 2a / y₁, giving −y₁ / 2a. The form below is point-slope with this normal slope; it is undefined at the vertex where y₁ = 0 (the normal there is the x-axis).

The slope form parameterises normals by slope m. Up to three normals of slope m can pass through a single external point, the basis of the theory of conormal points. The cubic dependence on m in the formula reflects that count.

Focal Distance

The focal distance is the distance from a point P(x₁, y₁) on the parabola to the focus. By the defining locus property, it equals the perpendicular distance from P to the directrix, which for y² = 4ax (directrix x = −a) is x₁ + a. The focal distance reads off the x-coordinate alone, no square roots required. It is non-negative because x₁ ≥ 0 on the parabola.

If P(x₁,y₁) lies on y²=4ax then focal distance = x₁+a

Angle Between Two Tangents From P(x₁,y₁) to Parabola y²=4ax is

From any external point exactly two tangents can be drawn to a parabola; the angle θ between them measures the opening of the curve seen from that point. The numerator radical is real precisely when (x₁, y₁) lies outside the parabola (y₁² > 4a x₁); on the curve the angle collapses to zero, and inside there are no real tangents. The denominator x₁ + a is the focal distance, so the formula relates the tangent angle to position relative to the directrix.