LOCUS

A geometrical shape (for example, a circle) was not thought of as an unlimited set of points until the beginning of the twentieth century; rather, it was thought of as an entity on which a point might be placed or moved. In the Euclidean plane, a circle is defined as the locus of a point at a specified distance from a fixed point, the circle’s Centre. Similar concepts are more typically formulated in modern mathematics by describing forms sets; for example, the circle is the set of points at a given distance from the Centre.

Consider the case of a moving point in a plane. The path taken by the moving point can be obtained by joining each of the positions through which it passes. If the point moves randomly, its path will be erratic, and we won’t be able to forecast, for example, how the point’s path would look when it comes to an end. As a result, it is impossible to predict the route of a randomly moving point. The ensuing path, on the other hand, will be predictable if the point’s movement follows a set of laws and can be a circle, a straight line, an ellipse, and so on.

A locus of points is a group of points whose location is determined by a set of rules. for example, A series of mountains in the northwest has been the site of several independence movements. The locus is depicted as the center of any location in this case.

A geometrical shape (for example, a curve) was not thought of as an endless set of points until the beginning of the twentieth century; rather, it was thought of as an entity on which a point might be placed or moved. In the Euclidean plane, a circle is defined as the locus of a point that is at a set distance from a fixed point, the circle’s center. Similar concepts are more typically restated in current mathematics by expressing forms as a set; for example, the circle is the set of points at a given distance from the center.

The set of all the points that fulfill some condition is generally called the locus of a point satisfying that condition. The usage of the singular in this formulation reflects the fact that mathematicians did not study infinite sets until the end of the nineteenth century. They saw lines and curves as places where a point may be located or moved, rather than as sets of points.

There are 6 basic theorems related to the locus that is discussed below: –

Questions about these locus theorems may include a construction component due to their connections to equal lengths, parallel lines, and angle bisectors.

Theorem 1

The locus at a constant distance “d” from the point “p” is treated as a circle, with “p” as the center and “d” as the diameter.

The region created by all points that are at the same distance from a single point can be determined using this theorem.

Theorem 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.

Theorem 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.

This theorem aids in the identification of the region formed by all points that are at the same distance from point A and point B. The perpendicular bisector of the line segment AB should be created.

Theorem 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.

Theorem 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.

Theorem 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.

Let us see some of the Examples of the locus in 2-D geometry: – Perpendicular bisector, Angle bisector, parabola, ellipse, hyperbola, circle etc. are examples of locus of some curve.

CONCLUSION: –

A set of points that satisfy a given attribute or condition is known as a locus of points. The circle on the map represents a collection of points that satisfy the property of being all 5 miles from our beginning point, forming a locus of points. Because the solutions to an equation are a set of points that satisfy the property of making the equation true, they turn out to be an example of a locus of points.