# Chapter 8 – Co-ordinate Geometry

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

Co-ordinate geometry is regarded as one of the fascinating mathematical ideas. Co-ordinate geometry (also known as analytic geometry) is a branch of mathematics that explains the relationship between geometry and algebra using graphs involving curves and lines. Geometric elements are given in Algebra, allowing students to solve geometric problems. It’s a form of geometry in which the positions of points on a plane are represented as ordered pairs of numbers. The basics of co-ordinate geometry (also known as Cartesian geometry) and its formulas and derivations are explained here.

#### Geometry of Co-ordinates

The analysis of geometry managing co-ordinate points is known as co-ordinate geometry (or analytic geometry). It means it’s attainable to obtain the length connecting a pair of points, divide lines in m:n ratios, find the middle point of a line, measure the specific areas of the given triangles in the Cartesian plane, and so on using co-ordinate geometry. Certain concepts of Cartesian geometry should be thoroughly comprehended.

#### What are a Co-ordinate and a Co-ordinate Plane?

You should feel comfortable plotting graphs on a plane using numbers from tables with linear and non-linear equations. The number line, also known as the Cartesian plane, is split into four quadrants by two perpendicular axes, the x-axis (horizontal line) and the y-axis (vertical line).

The four quadrants are mentioned below, along with their respective values:

• (+x, +y) in the first quadrant
• (-x, +y) is the second quadrant
• (-x, -y) is the third quadrant
• (+x, -y) is the fourth quadrant

The origin is defined as the point where the axes converge. A pair of values (x, y) expresses the position of some point on a plane, and these pairs are known as co-ordinates.

#### Cartesian System

Using two perpendicular lines, the co-ordinate system is used to locate a point’s direction in a plane. Points are defined in two dimensions as co-ordinates (x, y) with respect to the x and y-axes.

#### Ordinate and Abscissa

• The abscissa is a word that corresponds to a point’s perpendicular deviation from the y-axis determined along the x-axis.
• The ordinate is the y-coordinate of a position and is the perpendicular distance from the x-axis determined around the y-axis.

#### Solving Simultaneous Linear Equations

A simultaneous equation is one in which two or more quantities are connected using two or more equations. It consists of a limited number of separate equations. Simultaneous equations are also regarded as systems of equations because they are made up of a finite number of equations for which a general solution is found. To solve the equations, we must first decide the values of the variables concerned.

Simultaneous equations can be overcome in a variety of ways. Substitution, reduction, and the augmented matrix process are three standard methods of solving simultaneous equations. The two simplest methods that can efficiently solve the simultaneous equations to obtain correct solutions among these three methods are:

• Method of Elimination
• Method of Substitution

#### Elimination Method for Solving Linear Equations

The coefficients are multiplied with a constant in the elimination process. There are two broad classifications for solving a pair of linear equations in two variables in general. The graphical solution, for example, is a method of solving a pair of simultaneous equations by plotting the graphs for the specified equations.

#### Substitution Method for Solving Linear Equations

The algebraic approach for solving simultaneous linear equations is the substitution method. The value of one component from one equation is substituted in the other equation, as the name suggests. Therefore, a pair of linear equations are converted into a single linear equation of just one component, which can then be solved quickly.

#### Plotting of Points in the Plane for Given Coordinates

Any point can be located using a Cartesian co-ordinate system or Co-ordinate system, and that point can be plotted as an ordered pair (x, y) defined as Co-ordinates. The X-axis is the horizontal number line, and the Y-axis is the vertical number line; the point of convergence of these two axes is defined as the root, and the letter O denotes it.

• The two-dimensional plane is also regarded as the co-ordinate plane.
• The X-axis is labelled XX’, while the Y-axis is labelled YY.’

#### Conclusion

In this chapter, we learned about the concepts of co-ordinate geometry. We studied the Cartesian plane and also learned the method to plot points on a 2D plane.

#### FAQs

1. What are the advantages of utilising the substitution technique?
The replacement process has the advantage of having exact values for the variables (x and y) that refer to the intersection point.
2. Is it possible to solve a three-variable set of equations using the substitution method?
In general, when solving a three-variable set of equations, we may use either the replacement or reduction approaches to simplify the system to two equations of two variables.
3. In co-ordinate geometry, what are abscissa and ordinates?
On a line, the abscissa and ordinate indicate the direction of a node. The abscissa is the horizontal value on the X-axis, while the ordinate is the vertical value on the Y-axis.
4. What is a Cartesian plane, and how does it work?
A Cartesian plane comprises two perpendicular lines called the x-axis (vertical) and the y-axis (horizontal). The ordered pair can be used to calculate the precise location of a point in the Cartesian plane (x, y).
5. What are the applications of co-ordinate geometry?
Co-ordinate geometry is needed to provide a relation between algebra and geometry by using line and curve graphs.

Explore more about co-ordinate geometry through simple, interactive, and explanatory visualisations; download MSVgo – the Math Science Super App, which focuses on conceptual clarity.

### High School Physics

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### High School Math

• Algebra – Arithmatic Progressions
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### Middle School Science

• Acids, Bases And Salts
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### Middle School Math

• Area and Its Boundary
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• Data Handling
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• Similarity
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• Subtraction
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