Conics.docx

7/25/2019 Conics.docx

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Conics It is used to describe the diferent

curves ormed by a plane thatcuts through two pairs o

cones joined at their vertices.

Circle

ormed when a planeperpendicular to the

axis cuts through one o thecones

Ellipse

ormed when the plane cutsthrough one cone at an anglewith the axis

Parabola

ormed when the plane cutsonly a portion o the cones

Hyper bola

ormed when a plane parallel tothe axis cuts through the cones

GENERAL EQUATION of the o!r Conics"

Ax2 + y 2 + !x + "y + # $ %

Conics Ter#s Loc!s

$the locus o a curve is the set o points&'x( y) whose coordinates satisfy thegiven e*uation o the curve 'x( y) $%

%y##etry,-here is symmetry in a gure i all pointson one side o the axis o symmetry arere&ecte' on the other side.

'x( y) $ ',x( y) Intercepts

$ -hey are the points where the given curvecrosses the x axis and the y , axis$the vertices o ellipse( parabola andhyperbola

Asy#ptotes,-he line which the curve approaches butdoes not touch as it goes arther away

rom the origin


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CIRCLEA circle is the loc!s of

all points P ()* y+ ,hichha-e e.!al 'istances fro#a /)e' point

PROPERTIE% CENTER of the CIRCLE " the xed point

RA0IU% " the distance rom the center tothe circle

UNCTIONAL E.!ation "

Center " origin '%(% )

1 2 3 y 2 4 r 2

GENERAL E.!ation "

A) 2 3 5y 2 3 E 4 6 * where A45

Characteristics of the GENERALE.!ation"

78 oth variables x and y are *uadratic28 -he coe/cients o both variables are

e*ual98 oth terms have the same sign

Graph "

radius

c


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ELLIP%EAn ellipse is the loc!s of all

points P ()* y+ s!ch that the s!# of the 'istances of each point fro# thet,o /)e' points is constant

PROPERTIE%

a :b a ; b

%ol-in< forc

a 2 b 2 $ c 2 b 2 , a 2 $ c 2

!nctionalE.!ation

!enter 'h(0)

( x h)2

a2 + ( y k )

2

b2 = 1

!enter 'h(0)

( y k )2

b 2 + ( x h)

2

a 2 = 1

oci ' +c ( % ) ' % ( +c )

=ertices

x,intercept

'a(%)y,intercept'%(b)

x,intercept

'a(%)y,intercept'%(b)

Eccentricity

e = ca

e= cb

>a?or a)is x, axis y, axis

>inor a)is y, axis x, axis

Graph

GENERAL EQUATION

A) 2 3 5y 2 3 E 4 6* ,here A@5

Characteristics of the GENERALE.!ation"

1. oth x and y are *uadratic2. -heir coe/cients are not e*ual

. oth terms have the same sign

12

882

1

8

8


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PARA5OLAA parabola is the loc!s of all point P

( )* y + e.!i'istant fro# a /)e' point an' a/)e' line8

PROPERTIE%

UNCTIONAL

EQUATION

3 2$4ay

3 2$,4ay

y 2$ 4ax y 2$ 4ax

oc!s$ xedpoint

' %(a ) '%(,a) 'a(%) ',a(%)

0irectri), xed line y$ ,a y$a x$,a 3$a

=erte)$point onthe axishal waybetweenthe ocusand thedirectrix

' %(% ) ' %(% ) ' %(% ) ' %(% )

PrincipalA)is$the axis osymmetryo theparabola

y, axis 5,axis x,axis x,axis

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Conics.docx