SMN3023

ADVANCED CALCULUS

GROUP C

NAME MATRIX NUMBER

CHAN LEE LING D20092036897

ASMIDA BINTI CHE ME D20091034995

NOORUL ASMA BINTI ABDUL SAMAT D20092036907

SITI NAILAH SAKINAH BINTI MOHD SALEH D20091035018

WHAT DOES IT MEAN BY CONIC SECTION?

A conic section is the intersection of a plane and a cone.

Circle Ellipse(h) Parabola(h) Hyperbola(h)

Ellipse(v) Parabola(v) Hyperbola(v)

By changing the angle and location of intersection, we can

produce a circle, ellipse, parabola or hyperbola; or in the special

case when the plane touches the vertex: a point, line or 2

intersecting lines

Point Line Double Line

2 2 0Ax Bxy Cy Dx Ey F

The type of section can be found from the sign of:

then the curve is a...

< 0 ellipse, circle, point or no curve.

= 0 parabola, 2 parallel lines, 1 line or no curve

> 0 hyperbola or 2 intersecting lines.

2If 4 is.......b ac

General Equation for a Conic Section

TYPE OF CONIC SECTION

Circle Ellipse Hyperbola Parabola

FORMING OF CONIC SECTIONS

The conics get their name from the fact that

they can be formed by passing a plane

through a double-napped cone

CONIC SECTIONS : CIRCLE

DEFINITION OF CIRCLE

A circle is all points equidistant (the distance is called the radius)

from one point (which is called the center of the circle).

A circle can be formed by slicing a right circular cone with a plane traveling parallel to the base of

the cone.

RELATED TERMS IN CIRCLE

Arc: a curved line that is part of the circumference of a circle

Chord: a line segment within a circle that touches 2 points on the circle.

Circumference: the distance around the circle.

Diameter: the longest distance from one end of a circle to the other.

Origin: the center of the circle Pi: A number, 3.141592…, equal to

(the circumference) / (the diameter) of any circle.

Radius: distance from center of circle to any point on it.

Sector: is like a slice of pie (a circle wedge).

Tangent of circle: a line perpendicular to the radius that touches ONLY one point on the circle. 

HOW TO GRAPH A CIRCLE

Centre at the originCentre away from the origin

CENTRE AT THE ORIGIN Realize that the circle is centered at the

origin and place this point there. Calculate the radius by solving for r Plot the radius points on the coordinate

plane. Connect the dots to graph the circle using a

smooth, round curve.

CENTRE AWAY FROM THE ORIGIN

Locate the center of the circle from the equation (h, v).

Calculate the radius by solving for r. Plot the radius points on the coordinate

plane. Connect the dots to the graph of the circle

with a round, smooth curve

EXAMPLE

Show that the expression

represents the equation of a circle. Find its centre and radius.

SOLUTION Using completing the square

taking the free constants to the right-hand side

By comparing this with the standard form we conclude this represents the equation of a circle with centre at coordinate position (1, -3) and radius 2

The circle, as a wheel, is one of the greatest

inventions of all time and the basis of much of

our transportation system. Circular gears are

important elements in many of the machines we

use every day, from CD players to electric saws.

CIRCLES IN REAL LIFE APPLICATION

CIRCLES IN REAL LIFE APPLICATION

The distance around a circle is called the

circumference. The formula for finding the

circumference of a circle is "circumference of a

circle equals pi times the diameter of the circle":

CONIC SECTION : ELLIPSE

INTRODUCTION

An ellipse is a plane

curve that results from the

intersection of a cone by

a plane in a way that

produces a closed curve. 

THE PARTS OF THE ELLIPSE

An ellipse has two lengths (major

and minor axes)

Vertices of an ellipse are the

endpoints of the major axis

Foci of the ellipse always lie on the

major axis

THE EQUATION OF THE ELLIPSE

The point F1 and F2 are called

the foci (plural of focus) of the

ellipse.

The point P is a typical point

on the ellipse.

If we denote this constant by

2a, a > 0, then |PF1| + |PF2| =

2a for any point P on the

ellipse.

THE EQUATION OF THE ELLIPSE  d1 is the distance between (x,y)

and (-c,0) and d2 is the distance

between (x,y) and (c,0).

After substituting, we have our

desired equation:

HORIZONTAL ELLIPSE

Center: (0, 0)

Vertices: (a, 0) (-a, 0)

Foci: (c, 0) (-c, 0)

Major Axis: 2a

Minor Axis: 2b

Distance between foci:

2c

VERTICAL ELLIPSE

Center: (0, 0)

Vertices: (a, 0) (-a, 0)

Foci: (c, 0) (-c, 0)

Major Axis: 2a

Minor Axis: 2b

Distance between foci:

2c

TRANSLATION OF ELLIPSE

Center: (h, k)

Vertices: (a+h, k) (-a+h, k)

Foci: (c+h, k) (-c+h, k)

Major Axis: 2a

Minor Axis: 2b

Distance between foci:

2c

GEOMETRY OF ELLIPSE

The focal constant is equal to the

major axis.

Since the point (a, 0) is on the ellipse,

the sum of the distances from (a, 0) to

the foci (c, 0) and (-c, 0) equals the

focal constant. This distance is :

GEOMETRY OF ELLIPSE

Since the distance from (0, b)

to each focus is equal, the

distance from (0, b) to each

focus must equal a.

This creates a right triangle

with legs of length b and c, and

hypotenuse of length a, giving

the relation

APPLICATION OF ELLIPSE: SOLAR SYSTEM

The orbits of the planets are

ellipses, with the Sun at one focus of

the ellipse.

The orbits of the moon and of

artificial satellites of the earth are

also elliptical as are the paths of

comets in permanent orbit around

the sun

APPLICATION OF ELLIPSE: ELLIPTICAL GEAR

An ellipse is defined by a set of

points in a plane

This enables elliptical gears cut

about their foci to run at a constant

center distance.

Using precision elliptical bilobe

gears, flowmeters can have good

linearity over a wide range of flow

rates and viscosities.

CONIC SECTIONS : HYPERBOLA

DEFINITION•A hyperbola is the set of all points (x, y) such that the difference of the distances between (x, y) and two distinct points is a constant.

•The fixed points are called the foci of the hyperbola.

•The graph of a hyperbola has two parts, called branches.

• Each part resembles a parabola but is a slightly different shape.

•A hyperbola has two vertices that lie on an axis of symmetry called the transverse axis.

•For the hyperbolas, the transverse axis is either horizontal or vertical.

2 2

2 2

( ) ( )1 , where,

horizontal distance from the center to the box

vertical from the center to the box

distance from the center to the focus

= distance from the center to the vertex o

x h y k

a b

a

b

c

2 2 2

f box

For both vertical and horizontal hyperbolas,

a b c

EQUATION OF HYPERBOLA

GRAPH OF HYPERBOLA

Horizontal Tranverse AxisVertical Tranverse Axis

HORIZONTAL TRANVERSE AXIS

The branches of the hyperbola open left and right

2 2

2 2

( ) ( )1

x h y k

a b

VERTICAL TRANVERSE AXIS

2 2

2 2

( ) ( )1

y k x h

b a

The branches of the hyperbola open up and down.

ASYMPTOTE•The graph of an hyperbola gets fairly flat and straight when it gets far away from its center.

• From the graph, it will look very much like an "X", with maybe a little curviness near the middle.

•These "nearly straight" parts get very close to what are called the asymptotes of the hyperbola.

• For an hyperbola centered at (h, k) and having fixed values a and b, the asymptotes are given by the following equations:

( )b

y x h ka

( )a

y x h kb

Vertical

Horizontal

ECCENTRICITY•Hyperbolas can be fairly straight or else pretty bendy.

•The eccentricity is the ratio of the distance from the center to a focus divided by the distance from the center to a vertex.

•In other words, eccentricity can be defined as the measure of

the amount of curvature is the where

• Bigger values of e correspond to the straighter types of hyperbolas, while values closer to 1 correspond to hyperbolas whose graphs curve quickly away from their centers.

ce

a

EXAMPLE QUESTION

Find the center, vertices, foci, eccentricity,

and asymptotes of the hyperbola with the

given equation, and sketch the graph of

2 2

125 144

y x

SOLUTIONSince the y part of the equation is added, then the center, foci, and vertices will be above and below the center (on a line paralleling the y-axis), rather than side by side.

2 2

2 2 2

From the denominator, we know that

25 and 144

5 12

From equation , we know that 13

Then,

a b

a b

c a b c

ce

a

13

5e

2 2 2 2Since ( 0) and ( 0) then, the center is at ( , ) (0,0)

So, the foci are at (0, 13) and (0,13) and the vertices are at (0,5) and (0, 5)

x x y y h k

Because the y part of the equation is dominant (being added, not subtracted), then the slope of the asymptotes has the a on

top, so the slopes will be

5

12m

Center : (0,0)

Vertices : (0, 5) and (0,5)

Foci : (0,-13) and (0,13)

13Eccentricity,

55

Asymptotes, 12

e

y x

APPLICATION OF HYPERBOLA

•Sonic booms are created when an object exceeds the speed of sound in air

•The shock wave of a sonic boom takes the shape of a cone, and when it intersects the ground, it takes the shape of a hyperbola.

•Every point on the curve is hit at the same time, so everyone on the ground will hear the sound at the same time.

•Another application of hyperbolas involves radio waves. 

•When there are two points where radio signals are emitted, the signals form concentric circles intersecting each other.

•The patterns created by the intersecting circles of radio waves form the shapes of hyperbolas.

Definition :

Parabola is the set of all point P(x, y) in the plane that are equidistant from a fixed line L, called the directrix, and fixed point F, called the focus.

CONIC SECTIONS : PARABOLA

•Equation of a parabola

•Squaring both sides and simplifying lead to

x2 = 4py

•Standard form Standard form for the equation of a parabola with focus F(0,p) and directrix y = -p. In like manner, if the directrix and focus are x = -p and F(0,p), respectively, there find the standard form for the equation of the parabola is

y2 = 4px

2 2( )x y p y p

•Technique of graphing

vertex

F(o,p)fokus

y = -p directrix

axis

x2 = 4py, p > 0 x2 = 4py, p < 0

vertex

F(0,p)fokus

y = -p directrixaxis

y2 = 4px, p > 0

F(0,p)

fokus

x = -p

directrix

axis

vertex

y2 = 4px, p < 0

F(0,p)

fokus

x= -p

directrix

axis

vertex

Find the vertex, focus, axis, directrix, and graph of the parabola.

.

EXAMPLE QUESTION

2 4 8 28 0y y x

SOLUTIONIn order to write the equation in one of the standard forms we complete the square in y :

Comparing the last equation with (6) we conclude that the vertex is (-4,2) and that 4p=8 or p=2. From p=2>0, the parabola opens to the right and the focus is 2 units to the right of the vertex at (-2, 2).

The directrix is the vertical line 2 units to the left of the vertex, x = -6. Knowing that the parabola opens to the right from the point (-4,2) also tells us that the graph has intercept.

To find the x-intercept we set y=0 in (7) and find immediately 7

2x

2

2

4 4 8 28 4

( 2) 8( 4)

y y x

y x

The intercept is . To find the y-intercepts we set x= 0 in (7)

and find form the quadratic formula that or

and . The y-intercept are and

7( ,0)

2

2 4 2y 7.66y

3.66y (0,2 4 2)(0,2 4 2)

(-2, 2)

x = -6

(-4, 2)

(y - 2)2 = 8(x+4)

APPLICATION OF PARABOLA

1. Braking Distance Formula

The following table is taken from a Virginia Division of Motor Vehicles Manual and it shows the Reaction Distance, the Braking Distance, and theTotal Stopping Distance at various speeds.The Reaction Distance is the distance that your car travels from the time that the driver sees the need to do so until his foot hits the brake. The braking distance is the distance that the car travels after the brakes are applied until it comes to a stop.The Total Stopping Distance is the sum of the Reaction Distance and the Braking Distance.

2. Heater

Heaters are sold which make use of the relection property of the parabola. The heat source is at the focus and heat is concentrated in parallel rays.

 3. Path of a Ball

Gallileo was the first to show that the path of an object thrown in space is a parabola.

4. Antenna of a Radio Telescope   All incoming rays parallel to the axis of the parabola are reflected through the focus.

5. Flashlights & Headlights In terms of a car headlight, this property is used to reflect the light rays emanating from the focus of the parabola (where the actual light bulb is located) in parallel rays.

6. Parabolic Reflector

Parabolic reflectors work in much the same way as flashlights and antennas.

7. Path of a Projectile

Galileo Galilei found that all objects thrown form a parabolic path, no matter what. He deduced this by the simple observation of watching objects being thrown. Galileo is responsible for the modern concepts of velocity and acceleration to explain projectile motion that is studied today:A projectile which is carried by a uniform horizontal motion compounded with a naturally accelerated vertical motion describes a path which is a semi-parabola.