Definition of parabola

Definition of parabola:
A curve in a plane having two ends and one vertex is called a parabola. It is a conic section. That means it is a section of a double cone intercepted by a plane. When a double cone is cut by a plane such that the plane is parallel to any one slant length of the cone, the cross section thus obtained is called as such.

However in co-ordinate geometry we do not study parabola problems as sections of double cone. We use the characteristic property of these to define it and use that to find its equation.

It so happens that in the plane of every parabola there is a line (called the directrix) and a point (called the focus) such that for every point on this the distance from the directrix and the distance from the focus are found to be equal. This is the definition most commonly used in co-ordinate geometry to define these and to derive its equation.

Types of parabola:
Primarily these are of two types and they are horizontal and vertical. In simple terms, the horizontal form of this is x as a function of y. Therefore the equation of a horizontal parabola would be x = f(y) type. On the other hand in a vertical form of this the equation is y as a function of x. Therefore it would look like this: y = f(x).

Another way of classifying these would be: positive and negative parabola. For the horizontal form, if it opens towards the positive side of the x axis, it is called the positive para-bola; where as if it opens on the negative side of the x axis, it is called the negative para-bola. Similarly, for the vertical form, if it opens towards the positive y axis it is called a positive para-bola and if it opens towards the negative side of the y axis, it is called a negative form of same. Below are the pictures of the various types of these:

1. Horizontal positive form of the same:

As you can see in the above picture, this one opens to the right.
X = f(y) = p(y-k)^2 + h
2. Horizontal negative form:

The above form opens to the left.
X = f(y) = -p(y-k)^2 + h

3. Vertical positive form of the same:

A vertical positive form opens up (as can be seen in the picture above).
Y = f(x) = a(x-h)^2 + k

4. Vertical negative form:

Lastly the vertical negative form would open down.
Y = f(x) = -a(x-h)^2 + k