MATHEMATICS - FORM 2

LOCI IN TWO   DIMENSIONS

  

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 a child going down a slide is a line parallel to the slide.  

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

             

 Sketch and state the loci of the following moving objects. (a)   A pendulum that swings (b)   The blade of a moving 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 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 the locus.(a)   The locus of points with a constant distance from a fixed point O is a circle with centre O.

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

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

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

The locus of a point which is equidistant from T and V is a perpendicular bisector of the diagonal TV.  Therefore, the locus is the straight line UW that is another diagonal of the square TUVW.

(C)   Constructing the locus

To construct the locus:(i)   Describe and sketch the locus.(ii)  With appropriate scale, construct the locus using a 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 point O on a sheet of paper.  With point O as the centre, draw an arc 2 cm         from O to form a circle.

 Two 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. Steps to construct the locus :  1.   Draw a line segment TU of 5.4 cm.

  2.   Open the compasses to a radius more thatn half the length of 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 arcs to 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 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 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.

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

Line segment AB is 4 cm long.  A 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 at a constant distant of 1 cm from AB.  Fing the points of intersection of both loci.

  Step 1   Construct the loci of points M and N

  Step 1  Determine the points of intersection of both loci.

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

 

Object P is constantly moving at a distance of 3 units from the x-axis.  At the same time, it is also moving at a constant distance of 5 units from the origin.  Which of the points A, B and C shows the location of object P?

    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.

   The diagram shows a circle with a radius of 6 cm and centre O.  This circle is divided into 8 equal parts.  Which of the points P, Q, R, S, T, U, V and W are the points of intersection  between a locus which is 6 cm from O and a locus which is 6 cm from the straight line POT?

 Points P, Q, R, S, 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. 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.

Therefore, the intersections of these two loci are the points V and R.