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High School Mathematics
12.36 Introduction to Locus

Locus:
Locus is a Latin word which means location or place. The plural of locus is loci.

Definition:
A locus is a set of points satisfying a given condition or conditions.

Example:
Find the locus of points equidistant from a pair of parallel lines.
Solution:

Let 'l' and 'm' be two given parallel lines. Let us mark points A, B, C and D which are equidistant form both the parallel lines l and m. As we go on adding more and more points it will become increasingly clear that a pattern is emerging. The pattern emerging is a line. The pattern formed by all points which have the common property is called the locus of the points.

The locus of points that are equidistant from two parallel lines l and m is another line n parallel to l and m and laying between them.

Note:
Every point belonging to n is at a distance d(d > 0) from both l and m and that every point in the same place at a distance of d from both l and m belongs to n.

Points equidistant from two given points:

Let us mark two points A and B 8cm a part. Locate points P1 and P2 each on either side of AB such that P1A = P1B = 6cm.
and P2A = P2B = 6cm. Locate Q1 and Q2, such that Q1A = Q1B = Q2A = Q2B = 8cm.

Observe the pattern this set of points Q1, P1, P2, Q2........... seem to form the emerging pattern seems to be a line draw a line parring through all of them. Let us name it as line l.
Let l intersect AB at O. Note that OA = OB. Since OA = OB the mid point of AB also belongs to the required locus. Note Q1OB = 90. So Q1O ^ AB. Let us take some other point R on l. Compare the lengths of RA and RB we can find that RA = RB.

From the above example we notice that
In order to establish the locus of a points equidistant from two given points A and B is the perpendicular bisector of AB.

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