**What is ****Parabola: **If the intersecting plane is parallel to the generator of the cone but cut only one nappe is called parabola. We have already stated that a conic section is a parabola if e = 1. Here e is the eccentricity.

We shall first derive an **equation of a parabola in standard form** and study its important properties.

If we take the focus of the parabola as F (a, 0),\;a > 0 and its directrix as line L whose equation is x = -a, then its equation becomes very simple. Let P(x, y) be a point on the parabola.

So, by definition

\frac{\left|PF\right|}{\left|PM\right|}=1 \;\;\;\;or\;\;\;\; \left|PF\right|=\left|PM\right|

Now,

\left|PM\right|=x+a\;\;\;\;\;\;\;\;(1)

\left|PF\right|=\sqrt{(x-a)^2+(y-0)^2}

Substituting into (1), we get

\sqrt{(x-a)^2+(y-0)^2}=x+a

(x-a)^2+(y)^2=(x+a)^2

y^2=(x+a)^2-(x-a)^2

y^2=4ax

which is standard equation of **Parabola**

**Some Definitions**:

what is parabola

- The line through the focus and perpendicular to the directrix is called
**axis of parabola**. In case of (2), the axis is y = 0. - The point where the axis meets the parabola is called
**vertex of parabola**. Clearly the equation (2) has vertex .A(0,0).. The line through A and perpendicular to the axis of the parabola has equation x = 0.. It meets the parabola at coincident points and so it is a tangent to the curve at A. - A line joining two distinct points on a parabola is called a
**chord of parabola**. A chord passing through the focus of a parabola is called a**focal chord of parabola**. The focal chord perpendicular to the axis of the parabola (1) is called**latusrectum of parabola**. It has an equation x = a and it intersects the curve at the points where

y^2 = 4a^2 \;\;\;\;\;or\;\;\;\;\; y+\pm 2a

Thus coordinates of the end points L and L’ of the latusrectum are

L(a ,2a )\;\;\;\;\;\; and\;\;\;\;\; L'(a ,- 2a ).

The **length of the latusrectum** is \left|LL'\right|=4a

- The point (at^2 , 2at) lies on the parabola [/katex]y^2=4ax[/katex] for any real t.

x = at^2 \;\;\;\;\;\;\;\;\;,y = 2at

are called **parametric equations of parabola** y^2 = 4ax