Definition:
In geometry, a locus (plural: loci) (Latin word for “place”, “location”) is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.
In another terms…Locus can be defined as the path traced by a point, which moves so as to satisfy certain given conditions such as equidistant from two given lines, equidistant from a given point etc.
Theorems based on symmetry:
Theorem 2: The locus of a point equidistant from two intersecting lines is the bisector of the angles between the lines. |
Given: Two straight lines intersecting at . Point is such that it is equidistant from (the two given lines)
To Prove: Locus of is the bisector of
(i) lies on the bisector of
(ii) Each point on the bisector of is equidistant from
Proof: Consider and
(i) (Given)
(right angles triangle)
is common
Therefore
Therefore lies on the angle bisector of
(ii) Conversely, if be any point on the angle bisector
Consider
(given)
is common
Therefore
Hence
Theorem 3: The set of points equidistant from two points is a perpendicular bisector to the line segment connecting the two points. |
Given: Two fixed points . is a point equidistant from at all times.
To Prove: Locus of moving point is perpendicular bisector of line
(i) lies on perpendicular bisector of and conversely
(ii) Every point on this perpendicular bisector is equidistant from points
Proof:
(i) Consider
(Given)
(Given)
is common
Therefore (S.S.S postulate)
Therefore corresponding angles are equal
Hence
Since
Hence proved that lies on perpendicular bisector of
(ii) Given is the perpendicular bisector of
Consider
is common
Therefore
Hence
Therefore proved that every point on a perpendicular bisector is equidistant from fixed points .
Notes:
The locus of a point in a plane at a fixed distance from a given point is the circumference of a circle with the fixed given point as the center of the circle and the distance as the radius. | |
The locus of a point equidistant from two given parallel lines is a line parallel to the given lines and is midway between them. | |
The locus of a point, which is at a given distance from a given line, is a pair of lines parallel to the given line and at the given distance from it. | |
The locus of all mid-points of all equal chords, in a circle, is the circumference of the circle concentric with the given circle and having radius equal to the distance of equal chords from the center. | |
The locus of mid-point of all parallel chords in a circle is the diameter of the circle which is perpendicular to the given parallel chords. | |
The locus of a point equidistant from two concentric circles is the circumference of the circle concentric with the given circle and midway between them. |