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Introduction: A locus is derived from the Latin word which means “place”, “location”. The locus can be defined as the collection of all points (usually a line, a line segment, a curve, or a surface) whose placement fulfils or is defined by one or more stated conditions in geometry.
In other terms, the collection of points that meet a condition is frequently referred to as the locus of a point satisfying this property. The usage of the singular in this formulation demonstrates that mathematicians did not examine infinite sets until the end of the nineteenth century. Instead of seeing lines and curves as collections of points, they saw them as locations where a point may be situated or moved.
To demonstrate that a geometric form is the right locus for a particular set of requirements, the proof is often divided into two stages:
Proof that all of the points that meet the criteria are on the specified form.
Proof that all of the points on the given form meet the requirements.
In two-dimensional geometry, the locus of points is a curve or a line. A locus of points is a group of points that share a property or condition. The circle on the map represents the set of points that satisfy the property of being all 5 miles from our beginning point, forming a locus of points. When we were planning our path, we probably didn’t understand we were building a mathematical collection of points. A point locus generally yields a curve or surface. In our trekking example, the locus of points 5 kilometres from our starting point resulted in a circle-shaped curve. Now, how are curves often represented algebraically? We are correct in believing that we employ an equation.
The solutions to an equation are an example of a locus of points since they are a collection of points that meet the property of making the equation true.
locus of a circle
A circle is the locus of the set of all points that are a fixed distance from a fixed point.
The fixed-point is referred to as the circle’s “centre” in this context.
The fixed-distance is known as the circle’s “radius.”
Consider a moving point in a plane. The path taken by the moving point may be obtained by connecting each of the spots through which it passes. If the point moves at random, its path will be irregular, and we will be unable to forecast, for example, how the point’s path will appear when it stops. As a result, the path of a randomly moving point cannot be predicted. If, on the other hand, the movement of the point follows a set of laws, the resulting route is predictable and can be a circle, a straight line, an ellipse, and so on.
So, a circle, a straight line, and so on may be thought of as a collection of points arranged according to a set of rules.
Example of Locus
A perpendicular bisector to the line segment connecting two points is a group of points equidistant from two points.
The angle bisector is the collection of points equidistant from two intersecting lines.
A circle is a group of points whose distance from a single point is constant (the radius).
The set of points equidistant between a fixed point (the focus) and a line is known as a parabola (the directrix).
The collection of points for which the absolute value of the difference between the distances to two specified foci is a constant is known as a hyperbola.
Ellipse: a collection of points for which the total of their distances to two given foci is a constant.
Theorems on Locus of points:
In geometry, there are six popular locus theorems (rules).
Questions about these locus theorems may have a construction component due to their ties to equal lengths, parallel lines, and angle bisectors.
Theorem 1: A circle with P as its centre and d as its radius is the locus at a constant distance, d, from point P.
Theorem 2: The locus at a set distance, d, from a line m is a pair of parallel lines lying on either side of m at a distance of d from the line m.
Theorem 3: The perpendicular bisector of the line segment connecting two locations A and B is the locus equidistant from both.
Theorem 4: The locus equidistant between two parallel lines, m1 and m2, is a line that is parallel to both and halfway between them.
Theorem 5: The bisector of an angle is the locus in the interior of the angle that is equidistant from its sides (exclusive of the vertex).
Theorem 6: The pair of lines that bisects the angles generated by the lines m1 and m2 is the locus equidistant from two intersecting lines, m1 and m2.
Conclusion:
Locus a curve or other shape created by all points meeting a given equation of the coordinate relation or by a point, line, or moving surface. All forms, including circles, ellipses, parabolas, and hyperbolas, are defined by the locus as a set of points.
