# Parabola: Types of Parabolas Equations

## 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 : y^{2 } =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 ⇒ PF^{2} = PM^{2}

⇒ (x-a)^{2 }+y^{2} =(x+a)^{2}

⇒ y^{2} = (x + a )^{2} – (x+a)^{2}

⇒ y^{2} = (x + a )^{2} – (x- a)^{2}

⇒ y^{2 } =4ax (a > 0).

### 2. 2nd Standard Equation : y^{2 } =-4ax, a > 0

y^{2 } =-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 : x^{2 } =4ay, a > 0

x^{2 } =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 : x^{2 } = -4ay, a > 0

x^{2 } =-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