What is Parabola,Definition,Graph -

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