Understanding on Locus

Locus is a  set of points in a plane which satisfy certain geometrical conditions. The slope of any segment connecting any two points on this locus is constant, which is a distinctive trait of this locus. Basically Locus is related to Shape or Curve. We all are familiar with plotting of points in a plane, distance between two points, section formulae and various other basic concepts of coordinate geometry. So now we shall look at the topic Understanding the Locus.

Locus of Points

In geometry, a shape is defined by the locus of points. Assume that a circle is the location of all equidistant points from the centre. Similarly, the locus of the points defines additional forms such as an ellipse, parabola, hyperbola, and so on. Only curved shapes have a locus defined. These forms might be regular or asymmetrical. The forms with vertex or angles inside them are not described as loci. 

Locus of Circle

The locus is the collection of all points that satisfy the requirements and create geometrical shapes such as a line, a line segment, a circle, a curve, and so on. So, rather than perceiving them as a collection of points, we might think of them as locations where the point can be found or moved. The set of all points equidistant from a fixed point, where the fixed point is the circle’s centre and the distance between the sets of points is the radius, is known as the locus of the points or loci. 

Locus Theorem

There are main Six Locus Theorem  which are as follows:-

1. The locus at a constant distance “d” from the point “p” is treated as a circle, with “p” as the centre and “d” as the diameter. This theorem can be used to find the region formed by all points that are at the same distance from a single point.
2. A pair of parallel lines lying on either side of “m” at a distance “d” from the line “m” is considered the locus at a fixed distance “d” from the line “m.” This theorem aids in the discovery of the region created by all points that are at the same distance from a single line.
3. Perpendicular bisectors of the line segment that connects the two locations are loci that are equidistant from the two specified points, say A and B. The zone created by all points that are at the same distance from point A and point B can be identified using this theorem. The line segment AB’s perpendicular bisector should be drawn.
4. The locus that is equidistant between two parallel lines, say m1 and m2, is regarded as a line parallel to both m1 and m2 and should be midway between them. This theorem aids in the identification of the region defined by all points that are at the same distance from two parallel lines.
5. The bisector of an angle is the locus that is present on the interior of an angle and is equidistant from the sides of the angle.

The region created by all points that are at the same distance from both sides of an angle can be determined using this theorem. The angle bisector should be the region.

6. The locus is a pair of lines that bisects the angle formed by the two intersecting lines, say m1 and m2. This theorem aids in the identification of the region created by all points that are at the same distance from two crossing lines. The formed region should be a pair of lines that bisect the formed angle.

Examples of Locus in Two Dimensional Geometry

1. Angle Bisector :- Angle bisector is a locus or set of points that bisects an angle and is equidistant from two intersecting lines that create an angle.
2. Parabola :- A parabola is a group of points or loci that are equidistant from a fixed point and a line. The focus is the fixed point, and the line is the parabola’s directrix.
3. Perpendicular Bisector :- The perpendicular bisector is a set of points that bisects a line generated by linking two points and are equidistant from two points.
4. Ellipse:- Ellipse is defined as a set of points that satisfy the requirement that the sum of two foci point distances is constant.
5. Hyperbola:-  Two focal points are equidistant from the centre of the semi-major axis of a hyperbola. The collection of points that satisfy the requirement that the absolute value of the difference between the distances to two specified foci is a constant is known as a hyperbola.

 Steps to find Locus

1. Assume any random point P(a , b) on the Locus.
2. Write the equation and simplify it to get Locus.

Conclusion

A circle is the location of all points that are at a given distance from a fixed point. The perpendicular bisector of the line segment connecting the supplied two locations is the locus of all the points that are equidistant from them. The angular bisector of the angle created by two intersecting lines is the locus of all locations that are equidistant from the crossing lines. Locus is fixed point and it lies on the line segment of any 2 dimensional shape.