Locus

In Mathematics, locus meaning is a curve shape formed by all the points satisfying a specific equation of the relation between the coordinates, or by a point, line, or moving surface. Every shape such as circle, ellipse, parabola, hyperbola, etc. is represented by the locus as a collection of points.

You must have heard about the word location in real-life. The word location is derived from the locus itself. Locus represents the position of something. When an object is placed somewhere or something at a place is defined by the locus. For example, the area has become a locus of resistance to the point.

Locus Definition

A locus is a collection of points whose position is represented by certain conditions. For example, a range of the southwest has been the locus of several independence movements. Here the locus is represented as the center of any location.

In Mathematics, a locus is defined as the collection of points stated by a specific rule or law of equation.

Mathematics defines a locus as a curve or other shape made by points satisfying various equations of the relationship between coordinates, or by a point, line, or moving surface. All shapes including circles, ellipses, parabolas, hyperbolas, etc. arise from the locus as a set of points.

The word 'location' is something you already know in real life. Developed from locus itself, the word location is a derivative. The word locus simply refers to the position of something. Locates describe objects that are located in a place or events that take place there. Local opposition to the government has developed in the area, for instance.

Locus is What it Sounds Like

An axis of determination of a locus is all the points whose positions are determined by certain conditions. Several Independence movements have been led within a range of the Southwest. This definition of locus is used to determine the geographic center.

An equation or rule can be thought of as a locus, which is a set of points according to it.

Points of Convergence

In geometry, a shape is defined by the locus of points. Imagine that a circle is a point equidistant from all points that surround it. A point is defined by an ellipse or a parabola hyperbola, etc. In the same way, the locus of an ellipse is defined by a point.

Curves are the only shapes for which the locus is defined. Similarly, irregular shapes are also defined.

Shapes with vertex or angle inside are not described as loci.

Circumference of a Circle

A locus is a collection of points that form geometric shapes like lines, lines segments, circles, curves, etc., and have locations that satisfy the conditions for their location. The points can be viewed instead as places where a point can be located or moved instead of just as a set of points.

In terms of the locus of points or loci, the circle corresponds to all points that are equidistant from one fixed point, where the fixed point is the circle's center and its radius is the set of points from the center. Let P denote the point at the center of the circle, and r the radius of the circle, that is, the distance from P to the set of all points or the locus of all points.

What is the Meaning of Locus of a Point?

The locus of a point represents the shape in geometry. Suppose, a circle is the locus of every point which is equally distant from the center. Similarly, the other shapes such as an ellipse, parabola, hyperbola, etc are represented by the locus of points.

The locus is defined only for curved shapes. These shapes can be either regular or irregular.

Locus is exhibited for the shapes having vertex or angle between them.

Locus of a Circle

The locus of a circle is the collection of all points which form geometrical shapes such as line, a line segment, circle, a curve, etc., and whose location agrees the condition is the locus. Generally, we can say that instead of examining them as a collection of points, they can be seen as places where the points can situate or move.

In terms of the locus of point or loci, the locus of a circle is represented as the collection of all points which are equally distant from a fixed point.,where the fixed point is the center of the circle and the distance of the collection of points is from the center is the radius of the circle. Let us take m as the center of the circle and r is the radius of the circle, from the point, the collection of all the points, or to the locus of the points.

Locus Examples in Two-Dimensional Geometry

Here are some of the locus examples in two-dimensional geometry:

Perpendicular Bisector

The collection of points which bisect the line, molded by joining two points and equally distant from two points is called perpendicular bisector.

Angle Bisector

A locus of collection of points that bisect an angle and are equally distant from two intersecting lines, which forms an angle is known as an angle bisector.

Ellipse

The ellipse is defined as a collection of points that fulfill the condition where the sum of the distances of two focal points is fixed.

Parabola

It is the collection of points that are equally distant from a fixed point and a line is known as a parabola. The fixed point is represented as the locus and the line is represented as the directrix of the parabola.

Hyperbola

A hyperbola has two distinct focal points which are equally distant from the center of the semi-major axis. The collection of points fulfills the condition where the absolute value of the difference between the distances to two given foci is constant.

Solved Examples

1. The sum of the intercept cut off from the axes of coordinates by a variable straight line is 10 units. Find the locus of the point which divides internally the part of the straight line intercepted between the axes of coordinates in the ratio 2 : 3.

Solution: Let us assume that the variable straight line at any position intersects the x-axis at A (a, 0) and the y-axis at B (0, b).

clearly, AB is the part of the line intercepted between the co-ordinates axes. Further assume that the point (h, k) divides the line-segment AB internally in the ratio 2 : 3. Then we have,

H = (2 · 0 + 3 · a)/(2 + 3)

or, 3a = 5h

or, a = 5h/3

And k = (2 · b + 3 · a)/(2 + 3)

or, 2b = 5k

or, b = 5k/2

Now, by problem,

A + b = 10

or, 5h/3 + 5k/2 = 10

or, 2h + 3k = 12

Therefore, the required equation to the locus of (h, k) is 2x + 3y = 12.

2. Find the equation of the locus of a point P, the square whose distance from the origin is 4 times its y coordinate.

Solution: Let the given origin be A ( 2,0)

Let the point on the locus be P ( x,y)

The distance of P from X-axis = y

It is given that OP² = 4 PM

((X−0)²+(Y−0)2)²= (4y)² 

x² + y² - 4y = 0

Hence, the equation of the locus of P (x,y) is

x² + y² - 4y = 0

An Overview of Six Important Locus Theorems

Locus theorems are six dynamical theorems that are popular in geometry. Though these theorems may seem difficult, their concepts are actually simple to understand once you know what they are about. The following are six important theorems to understand.

1. The Location of the Focus:

The space formed by all of the points with the same distance from a single point can be found by applying this theorem

(Locus Theorem no.2):

By using this theorem, we are able to find the region formed by all points that are situated on a single line at the same distance from it. 

The Locus Theorem 3 States:

A theorem like this one can be used to determine the region formed by all the points which are located equally far from both point A and point B. An orthogonal bisector of line segment AB is formed by the perpendicular bisector of the line segment.

Hypothesis 4:

With this theorem, you can find the region bounded by two parallel lines and all the points that are at the same distance from them.

Locate Theorem 5:

An angle's region is defined by all the points located at the same distance from both sides as a result of this theorem. This area would be the bisector of the angle.

The Locus Theorem States:

A region is formed by all the points that are at the same distance from two intersecting lines if they are located at the same distance from them. This region is formed by a set of lines that divide the formed angle.

Conclusion

Hence,  Locus has been briefly explained with their distinctive theorems, how to solve the questions related to Locus, the Basic hypothesis regarding the locus, and their examples which make the topic keen to be learned.