A Detailed Overview on Locus

A locus is a curve or other shape that can be made in mathematics by all of the points that meet a given equation describing the relation between the coordinates, or by a point, line, or moving surface. The locus, which is a group of points, is used to define all of the shapes, including the circle, ellipse, parabola, and hyperbola, among others.

You must be familiar with the concept of “location” from your experience in the actual world. The root term, locus, ultimately gave rise to the English word “location.” The term “locus” is used to refer to the location of something. The term “locus” is used to refer to when something is positioned somewhere or when something took place at a certain location. For instance, the region has evolved into a focal point for rebellion against the government.

Locus

A locus is a collection of all the points in space whose relative positions are determined by a set of constraints. Take, for instance, a mountain range in the southwest of the United States that has played host to a number of independence movements. In this context, “locus” refers to the geographic centre of any given location.

A set of points that are represented by a particular rule, law, or equation is referred to as a locus in mathematics.

Locus of Points

In geometry, the shape of an object is determined by the distribution of its points. Let’s say that a circle is the location of all the points that are the same distance from the centre as one another. In a similar manner, the locus of the points is what defines the various other forms, such as an ellipse, parabola, hyperbola, and so on.

Only when curved shapes are considered can the locus be specified. These forms can be regular or irregular, depending on your preference.

There is no description of locus for the shapes that have a vertex or an angle within them.

The locus of a Circle

The locus is defined as the collection of all points whose locations satisfy the requirements and which collectively form geometrical structures such as a line, a line segment, a circle, a curve, etc. Therefore, we might say that rather than viewing them as a collection of points, we should view them as locations within which the point can be situated or moved.

The circle is defined in terms of the locus of the points, also known as the loci, as the set of all points that are the same distance from a fixed point. This fixed point is known as the circle’s centre, and the distance that each set of points is from the circle’s centre is referred to as the circle’s radius. Let us assume that point P is the centre of the circle and that the radius of the circle, denoted by the variable r, is the distance from point P to the collection of all points, also known as the locus of the points.

Locus theorems

There are six significant locus theorems.

In geometry, there are a total of six significant locus theorems that are commonly used. Although the concepts underlying these theorems may appear difficult to grasp at first glance, this reading should clear things up. Let’s have an in-depth conversation about these six important theorems.

• First Locus Theorem:

The circle that has the point “p” as its centre and the fixed distance “d” as its diameter is considered to be the locus that is at the fixed distance “d” from the point “p.”

Using this theorem, one may more accurately identify the region that is created by all of the points that are situated at the same distance from a single point.

• Locus Theorem 2 states that

The locus that is believed to be a pair of parallel lines that are placed on either side of “m” at a distance “d” from the line “m” is considered to be a locus that is at a fixed distance “d” from the line “m.”

The use of this theorem makes it possible to determine the region that is created by all of the points that are situated at the same distance from a single line.

• Locus Theorem 3 states that

The loci that are regarded to be perpendicular bisectors of the line segment that unites the two points are those that are equidistant from both of the given locations, in this case A and B.

The application of this theorem will assist in locating the region that is created by all of the points that are situated at the same distance from point A as they are from point B. It is expected that the newly generated region will correspond to the perpendicular bisector of the line segment AB.

• Locus Theorem 4 states that

The point at which the two parallel lines, say m1 and m2, are separated by an equal distance is known as the locus, and it must lie exactly in the middle of the two lines to be considered a line that is parallel to both of them.

The use of this theorem makes it possible to locate the region that is created by all of the points that are situated at the same distance from the two lines that are parallel to one another.

• Locus Theorem 5 states that

The point inside an angle that is equidistant from both of the angle’s sides is considered to be the angle’s bisector. This point can be found on the angle’s interior.

The application of this theorem will assist in locating the region that is created by all of the points that are situated at the same distance from both sides of an angle. It seems appropriate for the region to serve as the angle bisector.

• Locus Theorem 6 states that

The locus that is believed to be a pair of lines that bisects the angle generated by the two intersecting lines m1 and m2 is the point that is considered to be equidistant from the two intersecting lines, say m1 and m2.

With the assistance of this theorem, one may determine the region that is created by all of the points that are situated at the same distance from the two lines that cross. The area that is constructed should consist of a pair of lines that are positioned to bisect the angle that is produced.

Conclusion

A locus is a curve or other shape that can be made in mathematics by all of the points that meet a given equation describing the relation between the coordinates, or by a point, line, or moving surface. The locus, which is a group of points, is used to define all of the shapes, including the circle, ellipse, parabola, and hyperbola, among others.There is no description of locus for the shapes that have a vertex or an angle within them.There are six significant locus theorems.In geometry, there are a total of six significant locus theorems that are commonly used.