What is Parabola?
A parabola is a set of all points in a plane that are equidistant from a fixed line and fixed point in the plane.
- A line through the focus and perpendicular to the diretrix is called the axis of the parabola.
- The point of intersection of the parabola with its axis is called the vertex of the parabola.
- In the adjoining figure, C is a parabola with focus F and the line DD, as its directrix.
If we take an arbitrary point P on the parabola and draw PM 
DD then by the definition of a parabola, we have PF=PM.
Different Types of Parabolas Equations
1. First Standard Equation : y2 =4ax, a > 0
Let X’ OX and YOY’ be the coordinate axes and let a > 0 be given.
Let us consider a parabola whose focus is F(a,0) and the directrix is the line DD’, whose equation is x+a =0.
Consider a parabola whose focus is F(a,0) and the directrix is the line DD’, whose equation is x+a =0.
Let P (x,y) be an arbitrary point on the parabola. Let PM DD’.
Then, by the definition of a parabola, we have PF=PM.
Now, PF=PM ⇒ PF2 = PM2
⇒ (x-a)2 +y2 =(x+a)2
⇒ y2 = (x + a )2 – (x+a)2
⇒ y2 = (x + a )2 – (x- a)2
⇒ y2 =4ax (a > 0).
2. 2nd Standard Equation : y2 =-4ax, a > 0
y2 =-4ax (a > 0) is a parabola where,
- focus is F(-a, 0)
- vertex is O(0,0)
- directix is the line x-a = 0
- axis is the line y= 0
- length of the latus rectum is 4a
- latus rectum is the line x = -a
3. Upward Parabola Equation OR 3rd Standard Equation : x2 =4ay, a > 0
x2 =4ay (a > 0) is a parabola where,
- focus is F(0, a)
- vertex is O(0,0)
- directix is the line y+a = 0,
- axis is the line x= 0
- length of the latus rectum is 4a
- latus rectum is the line y – a = 0
4.Downward Parabola Equation or 4th Standard Equation : x2 = -4ay, a > 0
x2 =-4ay (a > 0) is a parabola where,
- focus is F(0, -a)
- vertex is O(0,0)
- directix is the line y-a = 0,
- axis is the line x= 0
- length of the latus rectum is 4a
- latus rectum is the line y + a = 0