LOCI IN TWO DIMENSION form2

MATHEMATICS MATHEMA TICS - FORM 2

LOCII IN TWO DIMENSION LOC DIMENS IONS S

DIMENSIONAL LOCI (A) Describing and sketching the locus of a moving object A locus in two dimensions is the path taken by a set of points on a plane that satisfy the conditions given. Loci are the plural of locus.

For example : (a) The locus of of a child going down a slide is a line parallel to the slide. slide.

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(b) The locus locus of a spinning yoyo is a circle. circle.

Sketch and state the loci of the following moving objects. (a) A pendulum that swings (b) The blade blade of a moving moving windmill windmill

(a) The locus of a pendulum that swings is an arc of a circle.

(b) The locus of the blade of a moving windmill windmill is a circle.

(B) Determining the locus To determine the locus of a point that satisfy a given condition, mark the possible positions of the points. Then, join the set of points to obtain obtain the locus. (a) The locus of of points with with a constant distance from a fixed point point O is a circle with centre O.





(b) The locus of of points that are equidistant from two fixed points is the perpendicular bisector of bisector of the straight line joining the two fixed points.

(c) The locus of points with with a constant distance from a straight line is two parallel lines on either side and equidistant from the straight line.

(d) The locus of of points that are equidistant from two intersecting lines lines is the bisector of the angle between the two intersecting lines.

In the diagram above, TUVW TUVW is a square. Determine the locus of a point which which is equidistant from T and V and moves within the square.





(C) Constructing the locus To construct the locus: (i) Describe and sketch the locus. (ii) With appropriate scale, construct construct the locus using a ruler, ruler, a set square or pair of compasses.

Construct the locus of point M such that it is always 2 cm from a fixed point O.

The locus of point M is a circle with centre O and a distance of 2 cm from the centre O. Step to construct the locus: 1. Open a pair of compasses on a ruler to measure a radius of 2 cm. 2. Mark a fixed fixed point O on a sheet of paper. paper. With point O as the centre, draw an arc 2 cm from O to form a circle.

Two points points T and U are 5.4 cm apart. Construct the locus of point A that is equidistant from T and U.

The locus of point A is the perpendicular bisector of the line joining the points T and U.





2. Open the compasses compasses to a radius radius more thatn thatn half the length of TU. TU. With point T as the centre, draw an arc below and above the line.

3. With the same radius and point U as the centre, draw two two arcs to intersect intersect the first two arcs at C and D.

4. Draw a line through C and D. This is the locus of point A.





INTERSECTIONS OF TWO LOCI (A) Determining the Intersections of Two Loci

The above figure shows the loci of particle A and particle particle B. Points P and Q are two points of intersection of the two loci.

The above figure shows a square ABCD with sides 8 cm. A point P moves moves inside the square such that it is equidistant from A and C. Another point Q moves inside the square such that it is at a constant distance of 6 cm from A. Find the points that satify both conditions. conditions.







The locus of P is the perpendicular perpendicular bisector of diagonal AC. AC. The locus of Q is an arc with centre A and radius 6 cm. The points that satisfy both conditions conditions are the points of intersection of both loci, X and Y.

Line segment AB AB is 4 cm long. A point point M moves such that it is at a constant distance of 2 cm from the midpoint of AB. Another point N moves such that it is is at a constant distant of 1 cm from AB. Fing the points of intersection of both both loci.

Step 1 Construct the loci of points M and N





Therefore, points P, Q, R and S are intersections of the two loci.





Point A(3, 5) is the location of object P.

The locus of a point 3 units from the x-axis is the line y = 3. The locus of a point 5 units from the origin is a circle with a radius of 5 units.

Therefore, the location of object P is point C.





Points P, P, Q, R, S, T, T, U, V and W are on the locus which is 6 cm from O. Points W and S are on the locus which is 6 cm from straight line POT POT.. Distance of D from T

Therefore, the intersections of the loci are points Q and S.

A locus which is 6 cm from O is a circle with a radius of 6 cm. A locus which is 6 cm from the straight line POT is a pair of parallel lines.

LOCI IN TWO DIMENSION form2

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