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CHAPTER 9

LOCI IN TWO DIMENSIONS

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9.1. Two-dimensional loci

1. Locus means place in Latin. Whenwe talk about finding the locus of

an object we want to find the placeswhere the object might be.

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2. Definition of locus: The path of a

moving point that satisfies thecondition (s) given.

For example:

The locus of a swinging

pendulum is the arc PQ

with centre O.

O

P Q

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3. The locus of a moving pointcan be determined byconstructing the positions ofthe point that satisfies thecondition(s) given.

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a) The locus of a point which is at aconstant distance from a fixed point

is a circle.For example: Determine the locus ofa point K, where Kis always 2 cm

from point O.

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O

Locus of K

The locus of Kis always 2 cm frompoint O. It is at a constant distancefrom point O. The locus of Kis a circle.

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b) The locus of a moving point whichis at equidistant from two fixedpoints is the perpendicular bisectorof the line which joins these twofixed points.

For example:The distance between two points, Xand Yis 6 cm. Determine the locus

of point L which is always atequidistance from point Xand pointY.

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The locus of L is always at equidistance frompoint X and point Y. The locus of L is

perpendicular bisector of the line which joinspoint Xand point Y, and divides the line intotwo equal parts where XL = LY= 3 cm.

X Y3 cm 3 cm

Locus of L

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c) The locus of a point which is at a

constant distance from a straightline is a pair of parallel lines.

For example:UVis a straight line. Determine thelocus of a point Pwhich is at 2 cm

from UV.

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The locus of Pwhich is at 2 cm from UVis apair of parallel lines. The distance of the locus

Pis from UVis always 2 cm.

U V

2 cm

2 cm

Locus of P

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d) The locus of a moving point which

is at equidistance from twointersecting lines is the two anglesbisectors.

For example:

GHand IJintersect at O. Construct

the locus of a point Rwhich isequidistance from GHand IJ.

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The locus of Ris at equidistance from GHand

IJ. It is the two angle bisectors of the anglesformed by lines. The locus of Rdivides theangle equally where IOR= ROH and GOR=

ROI.

I H

J

Locus of R

G

O

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9.2. Intersection of two loci

The intersection of two loci is the

points that satisfies the condition ofthe two loci. The intersection can bedetermined by drawing the loci.

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For example:PQRis a straight line. Mark theintersection points that satisfy the

conditions of the locus Xand thelocus Y, where

(i) Xis at 2 cm from PQR.

(ii) Yis at 3 cm from point Q.

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The intersection points of the locus of Xand the locus of Y satisfy the conditionsof the two loci.

Locus of X

= points ofintersection

RQP

3 cm2 cm

Locus of Y

2 cm

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THE END OF PAGE

PREPARED BY:

NORASHIKIN BINTI MOHAMAD