Form

y² = 4ax

Y² = -4ax

x² = 4ay

x² = -4ay

Shape of the parabola

Vertex

(0, 0)

(0, 0)

(0, 0)

(0, 0)

Focus

(a, 0)

(-a, 0)

(0, a)

(0, -a)

Equation of Directrix

x = -a

x = a

y = -a

y = a

Equation of axis

y = 0

y = 0

x = 0

x = 0

Tangent at the vertex

x = 0

x = 0

y = 0

y = 0

Parametric Equation

x = at², y = 2at

x = -at², y = 2at

x = 2at, y = at²

x = 2at, y = -at²

Focal Distance of (x₁, y₁)

x₁ + a

-x₁ + a

y₁ + a

-y₁ + a

Equation of Tangent

(a) At the point (x₁, y₁)

yy₁ = 2a(x + x₁)

yy₁ = -2a(x + x₁)

xx₁ = 2a(y + y₁)

xx₁ = -2a(y + y₁)

(b) in terms of m

y = mx + a/m

y = mx - a/m

x = my + a/m

x = my - a/m

(c) in terms of t

yt = x + at²

yt = -x + at²

xt = y + at²

xt = -y + at²

Equation of Normal

(a) At the point

     (x₁, y₁)

y - y₁ = -y₁/2a (x - x₁)

y - y₁ = y₁/2a(x - x₁)

x - x₁ = - x₁/2a (y - y₁)

x - x₁ = -x₁/2a(y - y₁)

(b) in terms of m

y = mx - 2am - am³

y = mx +2am + am³

x = my -2am - am³

x = my +2am + am³

(c) in terms of t

y=-tx+2at+at³

y = tx + 2at + at³

x = -ty + 2at + at³

x = ty + 2at + at³