Standard Equation of Parabola
Consider the focus of the parabola as S(a, 0) and directrix be x + a = 0, and axis as xaxis.
Now according to the definition of the parabola for any point on the parabola, we must have SP = PM
â‡’ = PN + NM = x + a
â‡’ (x â€“ a)^{2 }+ y^{2} = (a + x)^{2}
â‡’ y^{2} = (a + x)^{2} â€“ (x â€“ a)^{2}
â‡’ y^{2} = 4ax
Vertex:

(0, 0)

Tangent at vertex:

x = 0

Equation of latus rectum:

x = a

Extremities of latus rectum:

P(a, 2a), Q(a, â€“ 2a)

Length of latus rectum:

4a

Focal distance (SP):

SP = PM = x + a

Parametric form:

x = at^{2} and y = 2at, where t is parameter

Other Standard Forms of Parabola
Equation of curve  y^{2} = â€“4ax 
Vertex  (0, 0) 
Focus  (â€“a, 0) 
Directrix  x â€“ a = 0 
Axis  y = 0 
Tangent at vertex  x = 0 
Parametric form  (â€“at^{2}, 2at) 
Equation of curve  x^{2} = 4ay 
Vertex  (0, 0) 
Focus  (0, a) 
Directrix  y + a = 0 
Axis  x = 0 
Tangent at vertex  y = 0 
Parametric form  (2at, at^{2}) 
Equation of curve  x^{2} = â€“4ay 
Vertex  (0, 0) 
Focus  (0, â€“a) 
Directrix  y â€“ a = 0 
Axis  x = 0 
Tangent at vertex  y = 0 
Parametric form  (2at, â€“at^{2}) 
Equation of parabola when vertex is (h, k) and axis is parallel to the xaxis.
(y â€“ k)^{2} = 4a(x â€“ h)
Equation of parabola when vertex is (h, k) and axis is parallel to the yaxis.
(x â€“ h)^{2} = 4a(y â€“ k)