**Introduction to study for conic sections exam :-**

Circles ,parabola ,hyperbola ,ellipse are parts of conic section .They are called so as they are formed when a plane intersects a right circular cone.The conic sections are differentiated by their eccentricity.The eccentricity is the measure how a conic section is deviating from circular path.Here we will study for conic sections exam.

There are three types of conic sections

i) Parabola

ii) Hyperbola

iii)Ellipse

Explanation to conic sections :-

**Parabola** :-

It is the type of conic section having eccentricity as "1" .

Picture of formation of parabola :-

Here we can see the formation of parabola as cone is intersected by the plane.

Parabola is set of points in a plane that are equidistant from a fixed point "F" called as focus and a fixed line directrix.The point where the parabola chenges it direction is called the vertex of parabola. The line which is passing through the vertex and focus of parabola is known as axis of parabola and diretrix is the line which is perpendicular to axis of parabola.Vertex is equidistant from focus and directrix of parabola.

Parabola :-

Consider the below parabola

In the above parabola we can see the points A & B,which are on theparabola, are equidiatant from focus and directrix of parabola.

The general equation of aparabola is Y ^{2} =4ax. where a is focal length.

**Ellipse** :

The eccentricity of an ellipse is less than "1".

Picture of formation of ellipse :-

Here we can see the formation of ellipse as cone is intersected by the plane.

Ellipse is set of points in aplane sum of whose distance from two fixed points F1 F2 is a constant.

Ellipse picture :-

Consider the following ellipse with focus f1 anf f2.

In the above figure we can see the point "P" is equidistant from F1 & F2.We cen observe Yello blue and black lines showing the points

which are equidistant from foci F1 & F2.

General equation of an ellipse is x^{2} /a^{2} + y^{2} /b^{2} =1 with center at (0,0) where x^{2} /a^{2}
+ y^{2} /b^{2} =1 with center at (0,0)

The value of c is sqrt(a^{2} + b^{2} ) and foci are(c.0) (-c,0).

**Hyperbola :-**

The eccentricity of an hyperbola is greater then "1".

Picture of formation of Hyperbola :-

Here we can see the formation of hyperbola as cone is intersected by the plane.

Hyperbola is set of points in a plane the difference of whose distance from two points is constant.

hyperbola picture :-

Consider the following hyperbola

Here we can see the hyperbola with two foci F1 & F2 .

general equation of hyperbola is x^{2} /a^{2} - y^{2} /b^{2} =1 with center at (0,0)

The value of c is sqrt(a^{2} + b^{2} )

My forthcoming post is on Definition of a Rectangle with example,and this topic Surface Area of a Cube Formula will give you more understanding about Math.

**Ex:1**

Given the equation of parabola Y^{2} = 16X find the focus of parabola

solution:

Comparing with standard form of equation

Y^{2} = 4aX

we get vertex of parabola as

(0,0)

**Ex 2 :**

equation of a hyperbola is

9x^{2} - 16y^{2} = 144

Find foci .

solution :

9x^{2} - 16y^{2} = 144

can be written as

x^{2} / 16 - y^{2} / 9 = 1

so a= 4 & b=3

so focus = sqrt( 9+16) = 5

So foci are (5,0) &(-5,0).

**Ex 3 :**

equation of a ellipse is

9x^{2} +16y^{2} = 144

Find foci .

solution :

9x^{2} - 16y^{2} = 144

can be written as

x^{2} / 16 + y^{2} / 9 = 1

so a= 4 & b=3

so focus = sqrt( 9+16) = 5

So foci are (5,0) &(-5,0).