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# 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 : 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)+y2 =(x+a)2

⇒ y2 = (x + a )2  – (x+a)2

⇒ y2 = (x + a )2  – (x- a)2

⇒ y =4ax (a > 0).

### 2. 2nd Standard Equation : y2  =-4ax, a > 0

y =-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 x =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 x =-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
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