3 Sample Loci Assignment Questions & Answers To Boost Your Knowledge

Question 1: Define the locus of a point

Question 2: Which are the three common types of loci?

1. Circle

If a point P moves in one plane so that its distance from a fixed-point O is constant, then its locus is a circle.

To draw a circle, compasses are set to the required constant distance. With the point of the compass at O (Fig. 15.1), the compass lead then traces out the required circle through P1, P2, P3, etc., where OP=OP2 = OP3 = R, the radius of the circle.



Fig. 15.1 Circle Fig                                               15.2 Parallel line

2. Parallel line

If a point P moves in one plane so that its perpendicular distance from a fixed line AB is constant, then its locus is a line parallel to AB.

To draw a parallel line (Fig. 15.2):

With centres on AB, strike several arcs of radius R equal to To draw a parallel line

Draw a common tangent to all these arcs. This is the required distance between AB and the parallel line. parallel line.

3. Perpendicular line

If a point P moves in one plane so that it is equidistant from two fixed points A and B, then its locus is a straight line perpendicular to AB.

To draw a perpendicular line (Fig. 15.3):

With centres at A and B, strike two arcs each of an arbitrary radius R, to intersect at P, on either side of AB.

2 With the same centres, strike further pairs of arcs with radii R2, R3,..., etc. and intersection points P2, P3.... on either side of AB. A straight line drawn through P1, P2, P3,..., etc. is the required line perpendicular to AB (and is also the bisector of AB).

Fig. 15.3 Perpendicular line                                     Fig. 15.4 Ellipse

Question 3: Explain in detail two methods of drawing an ellipse

Answer

Ellipse - method 1

If a point P moves in one plane so that its distance from a fixed point C and its perpendicular distance from a fixed line AB is always in the same ratio 1:n, where n is any number greater than 1, then the locus of the point is an ellipse.

To draw an ellipse (Fig. 15.4):

1. Taking a distance ratio of, say, 1:2 draw a line parallel to AB and at an arbitrary distance 2R, from it.
2. From centre C, strike an arc of radius R, to intersect this line at P₁.
3. Repeat steps 1 and 2 with radii R2, R3,..., etc. to give intersection points P2, P3,..., etc.
4. Join all the intersection points by a smooth curve. This curve is the required ellipse.

Ellipse-method 2

If a point P moves in one plane so that the sum of its distances from two fixed points A and B is constant, then its locus is again an ellipse. If a piece of string of total length equal to AP+ PB is fixed with its ends at A and B and is kept taut by a pencil held against it inside the loop so formed, moving the pencil will produce a locus which is an ellipse.

To draw an ellipse (Fig. 15.5):

1. For measuring purposes, draw a construction line of the total length equal to AP+PB (Fig. 15.5(a)).
2. Draw the two fixed points A and B (Fig. 15.5(b)).
3. From centre A, strike an arc with radius AP₁.
4. From the centre, B, strike an arc with a radius BP, measured from the construction line. This arc intersects the first one at P₁.
5. Repeat steps 3 and 4 with various distances and intersection points P2, P3,..., etc. 6 Join all the intersection points by a smooth curve. The resulting locus is an ellipse.