Locus

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Locus in Mathematics: Exploring Definitions and Equations

Comprehensive Definition, Description, Examples & RulesĀ

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Introduction to Locus

A Locus is an important term of mathematics as it is considered a curve or a different type of shape made by all the points joining a particular equation of the relation between its coordinates or through the point or the line or the moving surface. All these shapes, such as the ellipse, a circle, or a parabola, can be defined through the locus as a set of points.Ā

In mathematics, A locus is a particular set of points that can satisfy some given condition or property and is a plurality considered Loci. These are sets of points at a distance similar to another point and satisfy a particular given equation. Locus meaning in Hindi is also available online.

Significance in Mathematics

The Locus in maths or Loci has effective importance, and it includes:

• It is a type of curve or a shape that is made through a point satisfying a particular equation.Ā
• The locus of points is directly equidistant from a fixed point at the centre.Ā
• It consists of a perpendicular bisector, which is the locus of points that is equidistant from two fixed points, A and B.Ā

Defining Locus (Loci)

Locus definition or Loci definition in mathematics can be termed as a shape or a particular shape made through the points that satisfy a particular equation of the relation among the coordinates through another point or line moving through the surface. All types of shapes, such as a parabola, a circle, or a hyperbola, can be defined through the locus as a set of points. A Locus can be termed a Loci and is a commonly used mathematical term that you can relatively use.

The locus is a particular set of points that helps to determine and satisfy your particular equation of the relation among the coordinates and the different points of the shape. It is the primary reason for you to use a locus.

Equation of Locus

The equation of the locus or loci is different from a regular geometrical equation as you need to derive the shape equation according to certain properties. The process by which you can derive the locus equation is:

• First, you have to assume the coordinates of the moving point of the locus as (X1, Y1)
• Then, you are to apply the geometrical condition on the two selected moving points, which helps you to obtain the relation between these two points.Ā
• In your result, you should replace X1 with x and Y1 with y.Ā
• The equation that you obtain after this will be your locus equation.

Examples

Common Loci Circles: x2+y2=2(x+y)

Find the equation of a pointās locus so that the sum of its distances from (0, -1) and (0,1) is 3. What exactly does the equation mean?

Solution

Suppose P(x, y) be a point on the given locus.

It is given that

A = (0, ā1)

B = (0, 1)

From the conditions given,

Distance from Point P to A + Distance from Point P to B = 3

Hence, we can write it as

PA + PB = 3

By rearranging the above equation,

PA = 3 ā PB

Now, Squaring both sides of the equation,

PA2=(3āPB)2

PA2 = 9 + PB2 ā 6PB

x2+(y+1)2 = 9+x2+(yā1)2ā6PB

(y+1)2ā(yā1)2ā9=ā6PB

4yā9 = ā6PB

Again, squaring both sides,

16y2ā72y+81 = 36PB2

16y2ā72y+81 = 36(x2+(yā1)2)

16y2ā72y+81 = 36x2+36y2ā72y+36

36x2+20y2 = 45

The above equation is written as follows

(x2/a2)+(y2/b2)=1

Hence, the above equation defines a circle

Therefore, the equation of locus is,

36x2+20y2 = 45, which is a circle.

Common Loci Lines: Y = MX + C

Find the locus of a point which is at a distance of 5 units from A(4,-3).

Solution:

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

The given geometric condition is,

PA=5

ā[(x-4)2 +(y+3)2] = 5

Squaring on both sides,

(x-4)2 +(y+3)2=25

x2 +16-8x+y2+9+6y=25

x2+y2-8x+6y=0 is the equation of locus.

Common Loci Parabola: y2+2ax+2by+c=0

Find the locus of a point which is equidistant from the points A(-3,2) and B(0,4).

Solution:

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

The given geometric condition is,

PA=PB

ā[(x+3)2 +(y-2)2] = ā[(x-0)2 +(y-4)2]

Squaring on both sides,

(x+3)2 +(y-2)2 = (x-0)2 +(y-4)2

x2 +9+6x+y2+4-4y= x2 +y2+16-8y

Simplifying the above equation,

6x + 4y = 3, which is the required equation of locus.

Locus in Real-life Applications

The locus concept is important in real-world situations, and the special areas where it has great significance are:

• Engineering: It is the sector in which the locus has an important role to play as the calculation of curves and locus points is very important in civil engineering and buildings, so calculating it in that way becomes very effective. Engineering involves the calculation of the common loci.Ā
• Physics: Both maths and physics have a joint connection. The locus plays an important role in physics, and effective calculations are made through the locus. Many physics theoreticians have used locus and their theorems, and there is indeed an important part.Ā
• Architecture: The field of architecture involves the creation of buildings and bridges, which directly involves the usage of calculations of lines and shapes. Using the locus to calculate the differences and curves is important to calculate the exact length, which will help in architecture.

Variation and Extension of Locus

Various advanced topics are related to the locus and are important in calculating the locus. These topics are:

Polar Coordinates

One of the most effective variations of a locus is its polar coordinates, which is a coordinating system in which the position of a point is given by a radical distance from an origin and the angle you can measure counterclockwise from a horizontal line through a polar axis. These are further concepts of the locus, and calculating the line segment becomes much easier through this.Ā

Conic Sections

A Conic section is the locus of a point that moves so that the distance from a fixed point always bears a constant ratio to its distance from another line, all in the same plane.

Parametric Equations

The parametric equation is a primary extension of the locus where the extension is from the two horizontal lines, one on each side of the x-axis, and the distance of each horizontal line from the x-axis is represented in the form of ‘d’ units. The calculation of the locus of the shape becomes very easy by using the parametric equations.Ā

These are the three primary extensions of the locus, which extend the entire concept of locus, and using the locus of a shape becomes very easy while you keep in mind the three variations of the locus.

Challenges and Problem-solving with Locus

There are certain challenges or problems that you can face while using the locus of the equation. These are:

• Determining the locus of a point is very difficult as it involves a lot of calculations.Ā
• Using the proper coordinates for the calculation can take time and effort.
• Determining the dimension of the geometrical figure is a necessary part where you might face difficulty in calculating the locus of a point.Ā

The only way you can solve all these challenges and problems is to follow the rules of calculating the locus and form an equation in a manner that will help you to calculate.

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Key Takeaways

1. A locus is considered a curve or a shape that joins through an equation.

2. The plural form of A locus is Loci.

3. It consists of many variations, making it a very special part of mathematical calculation when it is involved in making architecture and Engineering calculations very easy.

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There are certain steps that you have to follow while determining the equation of A locus, and these are:

• Assume the coordinates of the moving point of the locus as (X1, Y1)
• Apply the geometrical condition on the two selected moving points, which helps you to obtain the relation between these two points.Ā
• In your result, you should replace X1 with x and Y1 with y.Ā
• The equation that you obtain after this will be your locus equation.

The Lokesh has a primary usage in real life, and the areas where you can use it are:

• Architecture
• Engineering
• Physics

There are specific extensions and variations of A locus that make it a much more innovative mathematical concept. These are:

• Parametric Equations
• Conic Sections
• Polar Coordinates

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