Study For Conic Sections Exam

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.

 

Explanation to 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  :-

conic sections

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

conic sections

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 :-

conic sections1

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.

conic sections1

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 x2 /a2   +  y2 /b2  =1 with center at (0,0) where x2 /a2   +  y2 /b2  =1 with center at (0,0)

The value of c is sqrt(a2  +  b2  ) 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

hyperbola

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

general equation of hyperbola is x2 /a2   -  y2 /b2  =1 with center at (0,0)

The value of c is sqrt(a2  +  b2  )

 

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.

 

examples to study for conic sections exam

 

Ex:1

Given the equation of parabola Y2 = 16X find the focus of parabola

solution:

Comparing with standard form of equation

Y2 = 4aX

we get vertex of parabola as

(0,0)

 

Ex 2 :

 equation of a hyperbola is

9x2 - 16y2 = 144

Find foci .

solution :

9x2 - 16y2 = 144

can be written as

x2 / 16 - y2 / 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

9x2 +16y2 = 144

Find foci .

solution :

9x2 - 16y2 = 144

can be written as

x2 / 16 + y2 / 9 = 1

so a= 4 & b=3

so focus = sqrt( 9+16) = 5

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