Locus

What is the Meaning of 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. are 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.


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 the 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 agress 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 bisects 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 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.


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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 line is represented as the directrix of the parabola.


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Hyperbola

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


Solved Examples

1. Find the equation of the locus of a point which is at a distance 5 from (-2,3) in the poq plane.


Solution: Let M ( -2,3) be the given point,


Let X ( p,q) be any point on the locus 


It is given that XM = 5 


\[\sqrt{(- 2 - p)^{2} + (3 - q)^{2}}\] = 5


\[\sqrt{4 + 4p + p^{2} + 9 - 6q + q^{2}}\] = 5 


\[\sqrt{p^{2} + q^{2} + 4p - 6q + 13}\] = 5


On squaring both the sides, we get


\[(\sqrt{p^{2} + q^{2} + 4p - 6q + 13})^{2}\] = (5)²


p² + q² + 4p - 6q + 13 = 25


p² + q² + 4p - 6q = 25 -13


p² + q² + 4p - 6q = 12


The equation of the locus X (p,q) is


p² + q² + 4p - 6q = 12


2. Find the equation of the locus of a point P, the square of the 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


\[(\sqrt{(X - 0)^{2} + (Y - 0)^{2})^{2}}\] = (4y)² 


x² + y² - 4y = 0


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


x² + y² - 4y = 0


Quiz Time

1. A  moving point P retains equidistant from two fixed points. The locus of M represents:

  1. A straight line

  2. A circle

  3. A parabola

  4. A pair of straight lines


2. If M represents a moving point in  rectangular coordinates plane such that the distance between M and origin is 4, then locus of M is

  1. A triangle

  2. A parabola

  3. A straight line

  4. A circle


3. If M is a moving point in the rectangle coordinate plane such that the distance between M and the point (16,10) is equal to 4, then the locus of M is a 

  1. A circle

  2. Parabola

  3. Straight line

  4. Square 

FAQ (Frequently Asked Questions)

1. Explain the Ellipse in Terms of the Locus.

An ellipse in terms of the locus is defined as the collection of all points in XY- plane, whose distance from two fixed points ( known as foci) adds up to a constant value.


A circle is also represented as an ellipse, where the foci are at the same point which is the centre of the circle.


Ellipse is represented by its two-axis i.e. major and minor axis on XY-plane. The major axis is the longest diameter of the circle passing through the center from one point to the other at the broader portion of the ellipse. Whereas the minor axis is the shortest diameter of the axis passing through the center at the smaller portion.


Equation ellipse

When the center of the ellipse is at the origin point ( 0,0) and the foci are placed on the x-axis and the y-axis, then the equation of an ellipse can be easily derived as:


x²/ p² + y²/q² = 1

2. Explain the Hyperbola in Terms of the Locus.

The parabola is represented as the locus of a point which moves so that it always has equal distance from a fixed point ( known as the focus) and a given line ( known as directrix).


In the following graph,

  • The focus of the parabola is placed at ( 0,p)

  • The directrix is represented as the line y = -p

  • The focal distance| P |( distance from the origin to the focus or distance from the origin to the focus. We represent it through absolute value because distance is positive).

  • The points ( x,y) indicated any point placed on the curve

  • The distance from any point (x,y) to the focus ( 0,p) is equivalent to the distance from ( x,y) to the directrix.

  • The axis of symmetry if this parabola is the y-axis.