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### High School Mathematics12.36 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 ÐQ1OB = 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.