Mathematics Project On Conic Sections by Divya

8/10/2019 Mathematics Project On Conic Sections by Divya

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MATHEMATICS PROJECT

DIVYA XI A


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CONIC SECTIONS


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HISTORY

The names parabola and hyperbola are given byAPOLLONIUS.These curves are in fact !no"n as#ONI# S$#TIONS or more commonly #ONI#S

because they can be obtained as intersections of a plane"ith a double napped right circular cone. These curveshave a very "ide range of application in fields such as

planetary motion design of telescopes and reflectors inflash lights and automobile headlights etc .


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SECTIONS OF ACONE

Let l be a xed vertical line a nd mbe a nother l ine i ntersecting it at axed point V and inclined to it at anangle .

V

l m

Suppose we rotate the line maround the line l in such a way thatthe angle remains constant.

Then the surface generated is adouble-napped right circular h ollowcone herein after r eferred as coneand extending indenitely far inboth directions.


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The p oint V is called the vertex;

The l ine l is t he a xis of the co ne.

The r otating line m is called a generatorof the con e. The ver tex separates the con einto two p arts called nappes.

If we take the intersection of a p lane w itha cone, the sect ion so o btained is cal led aconic section.

Thus, conic sections are t he cu rvesobtained by intersecting a right circularcone by a p lane.

Let b e the angle made by theintersecting plane with the vertical axis.The intersection of the p lane can take p lacein any axis.


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CONIC SECTION

When a ri ght circular cone is intersected by a p lane, the cu rves ob tainedare known as conic sections.

If the p lane w hich cuts the cone is parallel the generator, then the conicsection obtained iscalARABOLA.

When the p lane which cuts the cone is not parallel to the generator, thenthe conic section obtained iscalldanELLIPSE.

When the plane which cuts the cone is parallel to the axis, then the conicsection isPERBOLA .


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Conic SectionsWhen the plane cuts the nappe (other than the vertex) of the cone, wehave the following situations :

(a) When = 90, the sect ion is a

circle.

(b) When < < 90, the section isan ellipse.


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(c) When = ; the section is a parabola

(d) When 0 < ; the plane cutsthrough both the nappes and thecurves o f intersection is a hyperbola


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When the plane cuts at the vertex of the cone, we have thefollowing different cases:

(a) When < 90o, then thesection is a point

(b) When = , the plane contains agenerator of the cone and thesection is a straight line. It is t hedegenerated case of a parabola.


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CIRCLE


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C i r cl e

The xed point is called the centre of the ci rcle a nd thedistance from the centre to a point on the c ircle i s ca lled theradius of the ci rcle

A Circle is the set of from a xed point in the plane.


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Given C ( h, k) be t he cen tre a nd r t he ra dius of circle. Let P(x, y) be any point on the circle. Then, by the d enition, | CP | =

r . By the d istance formula,

(x-h) 2 + (y-k) 2 =

r2

EQUATION OF A CIRCLE


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PARABOLA


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P a r a b

o l a A parabola is

from a xed line an d a xed point (not on the line) in the plane.

The xed line is called thedirectrix of the parabola andthe xed point F is called the

focus .(Para eans bola means throwing, i.e.,the shape described when youthrow a ball in the a ir).

A line through the focusand perpendicular to thedirectrix is ca lled the a xis

of the parabola. The pointof intersection ofparabola with the axis iscalled the vertex of theparabola .


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STANDARD EQUATIONOF PARABOLA

Y

O

Y

X X

F (a,0)M

lLet F be the focus and l the

directrix.

Let FM be perpendicular to the

directrix a nd bisect FM at the pointO.

Produce MO to X.

By the denition of parabola, the

mid-point O is on the parabola andis ca lled the vertex of the parabola.Take O as origin, OX as x-axis and OY perpendicular t o it as the y-

axis.

Let the distance from the d irectrix to the focus be 2 a.


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Let P( x, y) be any point on the parabola such that PF = PB, ... (1)

where PB is perpendicular to l.

The coord inates of B are ( a , y).

By the distance formula, we havePF = ( x a) 2 + y 2 and PB = (x + a) 2

Since PF = PB, we have

( x a) 2 + y 2 = ( x + a ) 2

x 2

2ax + a 2

+ y 2

= x 2

+ 2ax + a 2

y 2 = 4ax ( a > 0). (2)

Then, the coordinates of the focus are (a, 0), and the equation ofthe directrix is x + a = 0.

Y

O

Y

X X

F (a,0)

x

=

- a

M

l

P(x,y)

(-a, y) B

Or

,

And so, P(x, y) lies on theparabola.


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LATUS RECTUM

Y

O

Y

X X

F (a, 0)M

l

Latus rectum of a p arabola is a line segment perpendicular t othe axis of the parabola, through the focus and whose endpoints lie on the p arabola.

To nd the Length of the latusrectum of the parabola y2 = 4a x.

By the denitin AF = AC.

But AC = FM = 2a Hence AF = 2a.

And since the aborespect to x-axis AF = FB and so

AB = Length of the

C

A

B


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PARABOLA OF THE TYPE, y 2 =-4ax , a>0CHARACTRISTICS

Focus (-a,0)Equation of the axis is y= 0Equation of the directrix is x-a= 0Length of the latus rectum = 4a units


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PARABOLA OF THE TYPE , x 2 =4ay , a>0CHARACTRISTICS

Focus (0,aEquation of the axis is x= 0Equation of the directrix is y!a= 0Length of the latus rectum = 4a units


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PARABOLA OF THE TYPE , x 2 =-4ay , a>0CHARACTRISTICS

Focus (0,-a)Equation of the axis is x= 0Equation of the directrix is y-a= 0Length of the latus rectum = 4a units


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ELLIPSE


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Ellipse An ellipse is the set of a plane, the sum of whose

distances from two xed points in the p lane is a constant.

The two xed points are called the foci (plural of focus) of the ellipse.

P 1P 2 P 3

FocusF 1

FocusF 2

P 1F 1 + P 1F 2 = P 2F 1 + P 2F 2 = P 3F 1 +P 3F 2


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The m id point of the line seg ment joining the foci is called thecentre of the ellipse.

The l ine seg ment through the foci of the e llipse i s cal led the

major axis.

The line segment through the centre and perpendicular t o themajor a xis is ca lled the minor a xis.

The en d points of the m ajor a xis are ca lled the ver tices of the

ellipse.

We denote the length of the major axis by 2 a, the l ength of theminor axis by 2b and the d istance between the foci by 2 ae (2 c).Thus the length of the semi major a xis is a and that of the semiminor a xis is b.


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STANDARD EQUATION OF THE ELLIPSE:-

LF1dF2bhfid0bhidifh


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Let F1 and F2 be the foci and 0 be the mid-point of theline ment F1 , F2. let 0 be t he origin and the line from0 through F1 be the positive s-axis and through F2 asthe n egative axis. Let the line hrough 0 perpendicularto the x-axis be t he y-axis.Letthe co ordinates o f F1 andF2 be F 1(ae,0) and F2(-ae,0). Let P(x,y) be a ny point onthe ellipse.Suchthatsum of the d istance from P to thetwo foci be 2a.

By denition of the ellipse,PF1+PF2= constantPF1+PF2=2a


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= "a

#quaring $oth the sides,x " !"xae!a " e" !y " =4a " -4a !y " !(x-ae " !y "

x " !"xae!a " e" =4a " -4a !x " -"xae!a " e"

4xae=4a " -4a4a(xe = 4a %a- & =a-xe

#quaring again,x " -"xae!a " e" !y " = a " -"xae!x " e"

(x " -x " e" !y " = a " -a " e"

( (


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x " ('-e " !y " = a " ('-e "

, here $"

= a"

('- e"

) d h l


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)ere *+ and are the latus rectumLet *F'= (say/herefore, (ae,#ince *(ae, lies on ,

$ " =a " ('-e "


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/herefore, length of the latus rectum = units

ECCENTRICITY:-

/he eccentricity of an elli se is the ratio of the distance from thecenter of the elli se to one of the foci and to one of the 1ertices of theelli se and is denoted $y 2e3.

Elliftht ( Mj


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Ellipse of the type, ( Majoraxis a long y-axis )

Vertex (0, a

Foci (0, ae

Length of the minoraxis ="a

Length of the latusrectum = units


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HYPERBOLA


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THE HYPERBOLA

The plane that intersects the cone isparallel to the axis of symmetry of

the cone.

HYPERBOLA:


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HYPERBOLA:-

* hy er$ola is the set of all the oints in a lane the differenceof hose distance from t o fixed oints in the lane is the

constant.


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/he t o fixed oints are called the foci of the hy er$ola./he mid- oint of the line segment 5oining the foci is

called the centre of the hy er$ola./he line through the foci is called the /6*7#VE6#E

*89# and the line through the centre and er endicular tothe trans1erse axis is called the :7;< */E *89#. /he oints at hich the hy er$ola intersects the

trans1erse axis are called the VE6/E8 of the hy er$ola.istance $et een t o foci ="ae. Length of the trans1erse

axis ="a and the length of the con5ugate axis ="$.

STANDARDEQUATIONOFHYPERBOLA:


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STANDARD EQUATION OF HYPERBOLA:-

Let F' and F" $e the foci 0 $e the mid oint of the line


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Let F' and F" $e the foci 0 $e the mid- oint of the linesegment F' ,F". Let 0 $e the origin and the line through 0,through F' $e the ositi1e x-axis and through F" $e the

negati1e x-axis. /he line through 0 er endicular to x-axis$e the y-axis. Let the coordinates of F' and F" $e F'(ae, 0and F"(-ae, 0 .Let >(x,y $e any oint on the hy er$ola, such that>F " ->F ' ="a

>F " ="a!>F '

(x!ae " !y " =4a " !4a ! (x-ae " !y "

x " !"xae!a " e" =4a " !4a !x " "xae!a " e"


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x ! xae!a e =4a !4a !x - xae!a e4xae-4a " =4a4a(xe-a =4a(xe-a " =(x-ae " !y "

x " e" -"xae!a " =x " -"xae!a " e" !y "

(x " e" -x " -y" =a " e" -a "

x " (e " -' -y " =a " (e " -'

, here $ " =a " (e " -'

HYPERBOLAOFTHETYPE


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HYPERBOLA OF THE TYPE

Vertices (0, aFoci (0, ae


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THANK YOU

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Thank you foryour kindattention. Thispower pointpresentation wascreated by Divya

of class X I A, K.V A.S.C Centre[S

Mathematics Project On Conic Sections by Divya

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