Chapter 1 Coordinate Geometry

5/23/2018 Chapter 1 Coordinate Geometry

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Centre For Foundation Studies

Department of Sciences and Engineering

Chapter 1

FHMM1034Mathematics III

1


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Topics1.1 The Cartesian Coordinate

1.2 The Straight Line

1.3 Shortest Distance from a Point to aStraight Line


2

1.4 Circle

1.5 Intercepts and intersections


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1.6 Parabola

Topics

1.7 Ellipse

1.8 Hyperbola1.9 Shifted Conics


3

1.10 Parametric Equations

1.11 Loci


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.

The CartesianCoordinate


4


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Cartesian Coordinate

y

0x

a

bP(a, b)


55


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Distance between 2 points

lane.coordinate

on the),(and),(pointsLet 2211 yxQyxP

y

Q(x2,y2)

2 1y y

2y


60 x

P(x1,y1)

6

2 1x x

2x1x

1yR(x2,y1)


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Distance between 2 points

The Pythagorean Theorem gives

DISTANCE FORMULA

2 2

2 1 2 1PQ x x y y= +

2 2

2 1 2 1( ) ( )x x y y= +


7

( ) ( )

1 1 2 2

2 2

2 1 2 1

The distance for ( , ) and ( , ) is

P x y Q x y

x x y y +

7


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Example 1

Which of the points P(1, 2) or Q(8, 9) is

closer the point A(5, 3) ?


8


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Example 2

pointsthreeare),3(and)1,4(,)5,2(25CBA

area.itsfindandtriangleangled-righta

isthatshowHence,.and,ofdistancetheFindplane.coordinateon the

ABCACBCAB


9


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Point Dividing Straight Line

pointjoininglinethedivides),(If yxR

then,:ratioin then erna y,po nan, 2211

yxyx

+=

+=

1212 , yy

yxx

x


10


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,externallylinethedivides),(If PQyxR

then,signsoppositehavewilland

=

=

1212 , yy

yxx

x


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Example 3

divideswhichpointtheofscoordinatetheFind R

internally(i)

2:5ratioin the)5,4(pointtheand)3,8(pointthejoininglinethe

Q

P


12

externally(ii)


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13

R divides PQ internally. R divides PQ externally.


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Mid-Point

and),(pointofpoint-midThe 11 yxP

are),(point 22 yxQ

++

2,

2

2121 yyxx


14


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Example

Show that the quadrilateral with vertices

(1, 2), (4, 4), (5, 9) and (2, 7) is a

paralelogram by proving that its two

diagonals bisect each other.

P Q R S


1515


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1.2

The Straight Line


16


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Straight Line

passinglinestraightaofslopegradient /The m

is),(and),(pointsthrough 2211 yxQyxP

2112 , xx

xx

yym

=


17


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The Equation of Straight Line

throughpassinglinestraightaofequationThe

,,2211

121

xxyy

xxyy

=


18


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The Equation of Straight Line

gradientwithlinestraightaofequationThe m

s,po ntet roug tpass ngan a

)( axmby =


19


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Example 4

passinglinestraighttheofequationtheFind

.,an,po ntset roug t KH


20


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Example 5

withlinestraighttheofequationtheFind

).5,1(pointsthroughtpassesthat32gradient


21


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Parallel and Perpendicular Lines

linestheofgradientstherepresentandLet 21 mm

.tanandtanThen,

axis.positivethely torespective,

2211

2121

==

mm

x

parallel.areandlinestwothe,If(i) 2121 llmm =


22

lar.perpendicu

areandlinestwothe,1If(ii) 2121 llmm =


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Example

(a) Find an equation of the line through

t e po nt , t at s para e to t e line 4 6 5 0.

x y+ + =


23

perpendicular to the line 4 6 5 0

an

x y+ + =

d passes through the origin.


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Angle Between 2 Lines

andlinestwoebetween thangleThe 21 ll

yv

21

12

1

tan

mm

mm

+

=


24

angle.obtuseanis,0tanIf(ii)

angle.acuteanis,0tanIf(i)

1l2l


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Example 6

linesstraightebetween thangletheFind

.02035and0843 =+=+ yxyx


25


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1.3

Shortest Distance

from a Point to a


26


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The Shortest Distance

),(pointafromdistanceshortestThe

=

khPd

22 ba

cbkahd

+

++=


27

ve).orve(eithersignsamehaveall

sexpressionthe,0linestraight

,

+++

=++

cbkah

cbyax

ii

ii


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Example 7

fromdistancelarperpendicutheFind

.0543linestraighttheto)4,2(pointthe

=+

yxP


28


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Example 8

ifdeterminediagram,ausingWithout

.042

linestraighttheofsidesametheonlie)2,(and),2(pointsthe 2121

=+ yx

QP


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1.4

Circle


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Circle

),(centrewith thecircleaofequationThe baC

sra usan r

222 )()( rbyax =+


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,

222 ryx =+


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Circle

:iscircleaofequationgeneralThe

02222 =++++ cfygxyx

and),(iscentreitswhere fg


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.isradiusits cfg +

E l 9


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Example 9

circletheofequationanFind

).5,2(centerand3radiuswith


33

E l 10


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Example 10

thehasthatcircletheofe uationanFind

diameter.theof

endpointstheas)6,5(and)8,1(point QP


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E l 11


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Example 11

ofgraphthat theShow

radius.andcentreitsfindandcircle,ais

09)34(2 22 =+++ xyyx


35


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Example 12

at thecentrewithcircletheofe uationtheFind

.632

linethefromcentretheofdistanceshortest

thetoequalradiushavingand)2,3(point

+= xy

36


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Example 13

.02042circleon the

22=++ yxyx

37

E l 14


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Example 14

)2,4(pointtheofdistancelarperpendicutheFind A

pointtheofscoordinatethealsoFind.42line

straightthetouching)2,4(centrewithcircleaof

equationthefindThen,.42linestraighttheto

=+

=+

yx

A

yx

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.Intercepts andintersections


39

I


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Thex-coordinates of the points where a graph

Intercepts

intersects thex-axis are called thex-intercepts ofthe graph andy = 0.

They-coordinates of the points where a graph


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intersects they-axis are called they-intercepts ofthe graph andx = 0.


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Intercepts


41

Sety =0 and solve forx Setx = 0 and solve fory

E l 15


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Find thexintercepts andyintercept of the

Example 15

equation .22 =xy


42

P i t f I t ti


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In general, the coordinate of the points of

Points of Intersection

intersection of two equations can be found bysolving the two equations simultaneously.

Each real solution gives a point of intersection.


43

T f P i t f I t ti


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(i) 2 distinct real roots

Types of Points of Intersection

(ii) 2 equal real roots(iii)No real roots


44

(i) (ii) (iii)

Example 16


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Example 16

Find the coordinates of all points of intersection

xxxyxyyyxxy65,2(ii)

033,32(i)23

22

++===+=+

e ween e curves n eac o e o ow ngcases.


45

Example 17


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Example 17

and242ofra hsSketch the x =+

,andofscoordinatethefindingWithout

.andon,intersectiofpointsmark theand

diagram,sameon the106

2

QP

QP

xxy +=


46

.ofpoint-midtheofscoordinatethefind PQ


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1.6

Parabola


47

Parabola


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Parabola

A parabola is the set of points in the plane

equ s an rom a xe po n ca e e ocus and a fixed line l (called the directrix).


48FHMM1034Mathematics III

48

F(0,a)

y = - a

a

a

Parabola


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Parabola

isparabolaaofequationgeneralThe

ayx 42 =

directrix.thecalledislinefixedthe

andfocusthecalledis),0(pointfixedThe

= ay

aF


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.0foronlyexists

andaxisaboutsymmetricisgraphThe

y

y

Parabola


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If a>0 opens upward If a


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Parabola

Parabola with vertical Parabola with horizontal

Equation :

Properties: Vertex V (0,0)

Focus F (0, a)Directrix y = - a

Equation:

Properties: Vertex V (0,0)

Focus F (a, 0)Directrix x = - a

ayx 42 = axy 42 =


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Parabola


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Parabola

The line segment that

run t roug t e ocusperpendicular to the

axis with endpoints on

the parabola is called

aa

2a


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e a us rec um an

its length is the focal

diameter.

,

x = -a

Example 18


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Example 18

a) Find the equation of the parabola with vertex

, an ocus , , an s etc ts grap .

b) Find the focus and directrix of the parabola

and sketch the graph.06 2 =+ yx


53

,

the parabola and sketch its graph.221 xy =


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1.7

Ellipse


54

Ellipse


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An ellipse is the set of all points in the plane the

Ellipse

sum o w ose s ances rom xe po n s.

These 2 fixed points are the foci of the ellipse.

y

P(x, y)


55

x0F1(-c,0) F2(c,0)

Ellipse


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Ellipse

isellipseanofequationgeneralThe

12

2

2

2

=+b

y

a

x

axisma orthecalledis2andGenerall >


56

axis.minorthecalledis2and

b

Ellipse


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Ellipse


57

Ellipse


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Ellipse

12

2

2

2

=+a

y

b

x1

2

2

2

2

=+b

y

a

xEquation: Equation:

0>> ba0>>

baa a

c 222 bac = c222 bac =

c c

Vertices: ( , 0)

Major axis: 2a

Minor axis: 2bFoci: ( ,0) ,

Vertices: ( 0, )

Major axis: 2a

Minor axis: 2bFoci: (0, ) ,


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a

e=

a

e=, ,

Ellipse


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Eccentricity of an ellipse, e

Ellipse

the ellipse is.

Eccentricity of an ellipse,

The eccentricit of ever elli se satisfies

.

a

ce=

.10


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a p e 9

representequationsfollowingtheofeachthatShow

84(ii)

42(i)

:e psean

22

22

=+

=+

yx

yx


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curves.sketch theandcase,eachpropertiestheallState

Example 20


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Example 20

a) Find the equation of an ellipse with its

vert ces are an t e oc are .

b) Find the equation of the ellipse with foci

and eccentricity .4

=e

, ,

)8,0(


61

Example 21


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Example 21

Find the equations of the tangents with gradient 2

,63222

=+ yxto the ellipse with equation andfind their points of intersection.


62


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1.8

Hyperbola


63

Hyperbola


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yp

A hyperbola is the set of all points in the plane,

points. These fixed points are the foci of thehyperbola.

y

P (x, y)


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x0 F2 (c, 0)F1 (-c, 0)

Hyperbola


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yp

ishyperbolaaofequationgeneralThe

12

2

2

2

=b

y

a

x

:thatNote


65

.forexistnotdoescurveThe

axa


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veorveeitherofvaluelar eFor

:Notice

x +

:i.e.

2

2

22

b

x

a

by

x


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.asymptotestheareHence, xa

by

a

=

Hyperbola


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yp


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12

2

2

2

=b

y

a

x

Hyperbola


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Equation : Equation:

yp

12

2

2

2

= yx

12

2

2

2

= xy

Vertices ( ,0)

Asymptotes

Foci ( ,0) ,

Vertices (0, )

Asymptotes

Foci (0, ) ,

a

xa

by =

a

xb

ay =

c 222 bac += c 222

bac +=


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The Rectangular Hyperbola


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:If ba =

toreduceshyperbolaaofequationThe

222ayx =

h erbolarrectan ulathecalledish erbolaThis


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.areasymptoteswith the

xy =

The Rectangular Hyperbola


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The graph of and

222ayx =

2cxy =

are shown below222

ayx =y

2

cxy =


70

x0

Example 22


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:curvesfollowingeSketch th

5(ii)

149(i)

22=

=

yx

yx


71

4)1((iii) =yx

Example 23


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a) State all the properties of the hyperbola and

b) Find the equation of the hyperbola withvertices and foci .

099(ii)144169(i) 2222 =+= yxyx

)0,4()0,3(


72

c) Find the equation of the hyperbola with

vertices and asymptotes .)2,0( 2=y


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1.9

Shifted Conics


73

Shifted Conics


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In previous section, we studied parabolas with

vert ces at t e or g n an e pse an yper o as

with centers at origin.

In this section, we consider conics whose


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origin, and we need to determine how this affectstheir equations.

Shifted Conics


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Given h and kare positive real numbers,

Replacement How the graph is shifted

1. x replaced byx h

2. x replaced byx + h

Right h units

Left h units


75

(Page 776)

.

4. y replaced byy + k

Downward kunits

Shifted Ellipses


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General equation of an ellipse:

If we shift it so that its center is at the point (h, k)instead of at the origin, then its equation

122 =+by

ax


76

becomes:1

)()(2

2

2

2

=

+

b

ky

a

hx

Shifted Ellipses


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77

(Page776)

Example 24


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Sketch the graph of the ellipse

and determine the center.

19

)2(

4

)1( 22=

+

+ yx


78

Shifted Parabolas


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Applying shifts to parabolas leads to the

equations and graphs shown as followings:


79

0for

)(4)()i( 2

>

=

a

kyahx

0for

)(4)()ii( 2

=

a

hxaky

0for

)(4)()iv( 2

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Chapter 1 Coordinate Geometry