Conic Section

A conic section or conic is defined as the locus of a point which moves in a plane so that its distance from a fixed point in the plane bears a constant ratio to its distance from a fixed line in the plane.  The fixed point is called focus, the fixed line is called the directrix, and the constant ratio is called eccentricity of the conic. The eccentricity of a conic is denoted by e.

When e = 1, the conic is called parabola.

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A figure of a parabola is given above.

Standard form of a parabola

The standard form of a parabola is given by y = ax2 + bx + c, a ≠ 0. It is the general quadratic equation.
If a is greater than 0, the parabola opens upwards and if a is less than 0, the parabola opens downwards.

Axis of symmetry of the parabola in standard form

The axis of symmetry of a parabola in standard form is given by x= -b/2a

Lets consider a few examples

Example 1: -
Write the equation of the parabola whose focus is S(-4,5) and vertex (-4,1).
Solution: -
Distance between the focus and the vertex = 4
Distance between the vertex and the directrix =  4
Distance between the vertex and x axis = 1
Therefore distance between x axis and directrix =3
Since the directrix is parallel to x axis and at a distance of 3 units below the x axis, the equation to the directrix is  y = -3

Let P(x,y) be any point on the parabola.  Let PM be the perpendicular to the directrix.

Then SP2 = (x + 4)2 + (y - 5)2
PM2 = (y + 3)2
Since SP2 = PM2, we get
(x + 4)2 + (y - 5)2 = (y + 3)2
Simplifying we get
x2 + 8x + 16 + y2 - 10y + 25 = y2 + 6y + 9
Cancelling common terms, we get
x2 + 8x + 16 - 10y + 25 = 6y + 9
subtracting 6y  and 9 on both sides, we get
x2 + 8x -16y +32 =0
Write all terms in x and constants on one side
16y = x2 + 8x + 32
Dividing by 16 on both sides, we get
y = x2/16 + 1/2x + 2, is the required equation of the parabola.

Writing numbers in the standard form

The standard form is the method of representing bigger numbers in a simple form. We write a number in the form x * 10y,where x is a decimal number with only one number before decimal.

Example 1: -
Write the number 56832 in the standard form.
Solution: -
Given number is 56832. This can be written as 56832.0
In the standard form we write the number in the form x * 10y,where x is a decimal number with only one number before decimal.
That is 56832.0 = 5.6832 x 104
So the standard form of 56832 = 5.6832 x 104

Example 2: -
Write the number 23.476 in the standard form.
Solution: -
Given number is 23.476. we have to write this number in standard form.
In the standard form we write the number in the form x * 10y,where x is a decimal number with only one number before decimal.
That is 23.476 = 2.3476 x 101
So the standard form of 23.476 = 2.3476 x 101

Example 3: -
Write the number 0.0032 in the standard form.
Solution: -
Given number is 0.0032. we have to write this number in standard form.
In the standard form we write the number in the form x * 10y,where x is a decimal number with only one number before decimal.
That is 0.0032 = 3.2 x 10-3
So the standard form of 0.0032 = 3.2 x 10-3