## Parabola

A parabola is a curve where any point is at an equal distance from a fixed point(focus) and a fixed line (directrix).

For example when we kick a ball, or throw a stone, or shoot an arrow the path we get by the ball or stone or arrow is a para-bola.

The important names involved are:

• Focus: Fixed point
• Directrix: Fixed line.
• Vertex:  It is the point where the curve changes the direction. It is in the middle of focus and directrix.
• Axis of symmetry: Passes through the focus, and at right angles to the directrix.
• Latus rectum: The chord which passes through the focus and is perpendicular to the axis is called the latus rectum.

Properties:

• Any ray parallel to the axis of symmetry gets reflected off the surface direct to the focus.
• When we slice through a cone,we get this shape.

Let S be the focus and the line DD′ be the directrix. Draw SA perpendicular DD′ cutting DD′ at A. Let SA = 2a. Take AS as the axis of X and OY perpendicular to AS through the middle point O of AS as the Y axis.

Then S the focus is (a,0) and directrix DD′ is the line x+a=0.

Let P(x,y) be any point on the curve. Draw PM perpendicular to DD′.

 PM= NA = NO+OA = x+aSP² = (x-a)² +y²Then SP/PM = e =1.           SP²     =      PM²  (x-a)² + y²    =     (x+a)²   x² +a²-2ax+y²=   x²+a²+2ax              y²    =     4axThis is the standard equation

Different forms:

1. y²   = 4ax Here the focus is (a,0)
2. y²   = -4ax, here the focus is (-a,0).
3. x²   = 4ay, here the focus is (0,a)
4. x²   = -4ay, here the focus is (0, -a).

Example:

Find the focus of y² = 3ax.

Solution:

Standard equation is y² = 4ax.

Converting the given equation in the standard form.

y²   = 4(3/4)ax.

Here 3/4 stands for a. so a = 3/4.

Focus = F = (a, 0) = (3/4, 0)

Related Topics

Parents and teachers can guide the students to go through the standard equation, its important properties, different forms and example. If you have any doubt you can contact us through mail, we will help you to clear your doubts.

Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.  They are: