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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.

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.

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.

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.

  • 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.

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

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.

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.

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.’

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.

  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

  • Alternating Current
  • Atoms
  • Communication Systems
  • Current Electricity
  • Dual nature of Radiation and Matter
  • Electric Charges and Fields
  • Electricity
  • Electromagnetic Induction
  • Electromagnetic Waves
  • Electron Beams and Radioactivity
  • Electrons and Photons
  • Electrostatic Potential and Capacitance
  • Fluid Pressure
  • Force and Acceleration
  • Force And Laws Of Motion
  • Gravitation
  • Internal Energy
  • Kinetic Theory
  • Law of motion
  • Light – Reflection And Refraction
  • Magnetic Effects Of Electric Current
  • Magnetism and Matter
  • Management Of Natural Resources
  • Mechanical properties of Fluids
  • Mechanical properties of Solids
  • Motion
  • Motion in a plane
  • Motion in a straight line
  • Moving Charges and Magnetism
  • Nuclear Energy
  • Nuclei
  • Oscillations
  • Our Environment
  • Paths of Heat
  • Physical world
  • Ray optics and optical instruments
  • Semiconductor Devices
  • Semiconductor Electronics: Materials, Devices and Simple Circuits
  • Simple Machines
  • Sound
  • Sources Of Energy
  • Specific and Latent Heats
  • Spherical Mirrors
  • Static Electricity
  • Systems of Particles and Rotational motion
  • Thermal properties of matter
  • Thermodynamics
  • Units and Measurement
  • Vectors, Scalar Quantities and Elementary Calculus
  • Wave Optics
  • Waves
  • Work, Power and Energy

High School Chemistry

  • Acids, Bases and Salts
  • Alcohols, Phenols and Ethers
  • Aldehydes, Ketones and Carboxylic Acids
  • Aliphatic and Aromatic Hydrocarbons
  • Alkyl and Aryl Halides
  • Amines
  • Analytical Chemistry 
  • Atomic Structure
  • Atoms And Molecules
  • Basic concepts of Chemistry
  • Biomolecules
  • Carbon And Its Compounds
  • Carboxylic acids and Acid Derivatives
  • Chemical Bonding and Molecular Structures
  • Chemical Energetics
  • Chemical Equilibria
  • Chemical Kinetics
  • Chemical Reactions And Equations
  • Chemical Reactions and Their Mechanisms
  • Chemistry in Everyday Life
  • Chemistry of p-Block elements
  • Chemistry of Transition and Inner Transition
  • Classification of Elements
  • Coordination Compounds
  • Cyanide, Isocyanide, Nitro compounds and Amines
  • Electrochemistry
  • Electrolysis
  • Elements, Compounds and Mixtures
  • Environmental Chemistry
  • Equilibrium
  • Ethers and Carbonyl compounds
  • Haloalkanes and Haloarenes
  • Hydrocarbons
  • Hydrogen
  • Ideal solutions
  • Introduction to Organic Chemistry
  • Ionic equilibria
  • Matter
  • Matter Around Us
  • Matter In Our Surroundings
  • Metallurgy
  • Metals And Non-Metals
  • Mole Concept and Stoichiometry
  • Natural Resources
  • Organic Chemistry – Basic Principles
  • Periodic Classification of Elements
  • Physical and Chemical Changes
  • Physical and Chemical Properties of Water
  • Polymers
  • Preparation, Properties and Uses of Compounds
  • Principles and Processes of Isolation of Elements
  • Redox Reactions
  • Relative Molecular Mass and Mole
  • States of Matter
  • Structure Of The Atom
  • Study of Compounds
  • Study of Gas Laws
  • Study of Representative Elements
  • Surface Chemistry
  • The d-block and f-block elements
  • The Gaseous State
  • The p-Block Elements
  • The Periodic Table
  • The s-Block Elements
  • The Solid State
  • Thermodynamics

High School Biology

  • Absorption and Movement of Water in Plants
  • Adolescent Issues
  • Anatomy of Flowering Plants
  • Animal Kingdom
  • Bacteria and Fungi-Friends and Foe
  • Biodiversity and Conservation
  • Biofertilizers
  • Biological Classification
  • Biomedical Engineering
  • Biomolecules
  • Biotechnology and its Applications
  • Biotic Community
  • Body Fluids and Circulation
  • Breathing and Exchange of Gases
  • Cell – Unit of Life
  • Cell Cycle and Cell Division
  • Cell Division and Structure of Chromosomes
  • Cell Reproduction
  • Cellular Respiration
  • Chemical Coordination and Integration
  • Circulation
  • Control And Coordination
  • Crop Improvement
  • Digestion and Absorption
  • Diversity In Living Organisms
  • Ecosystem
  • Environmental Issues
  • Excretory Products and their Elimination
  • Flowering Plants
  • Genes and Chromosomes
  • Health and Diseases
  • Health and Its Significance
  • Heredity And Evolution
  • Heredity and Variation
  • How Do Organisms Reproduce?
  • Human Diseases
  • Human Eye And Colourful World
  • Human Health and Disease
  • Human Population
  • Human Reproduction
  • Hygiene
  • Improvement In Food Resources
  • Integumentary System- Skin
  • Kingdom Fungi
  • Kingdom Monera
  • Kingdom Protista
  • Life Processes
  • Locomotion and Movement
  • Microbes in Human Welfare
  • Mineral Nutrition
  • Molecular Basis of Inheritance
  • Morphology of Flowering Plants
  • Neural Control And Coordination
  • Nutrition in Human Beings
  • Organism and Population
  • Photosynthesis
  • Photosynthesis in Higher Plants
  • Plant Growth and Development
  • Plant Kingdom
  • Pollination and Fertilization
  • Pollution; Sources and its effects
  • Principles of Inheritance and Variation
  • Reproduction and Development in Angiosperms
  • Reproduction in Organisms
  • Reproductive Health
  • Respiration in Human Beings
  • Respiration in Plants
  • Respiratory System
  • Sexual Reproduction in Flowering Plants
  • Strategies for Enhancement in Food Production
  • Structural Organisation in Animals
  • Structural Organisation of the Cell
  • The Endocrine System
  • The Fundamental Unit Of Life
  • The Living World
  • The Nervous System and Sense Organs
  • Tissues
  • Transpiration
  • Transport in Plants

High School Math

  • Algebra – Arithmatic Progressions
  • Algebra – Complex Numbers and Quadratic Equations
  • Algebra – Linear Inequalities
  • Algebra – Pair of Linear Equations in Two Variables
  • Algebra – Polynomials
  • Algebra – Principle of Mathematical Induction
  • Algebra – Quadratic Equations
  • Binomial Theorem
  • Calculus – Applications of Derivatives
  • Calculus – Applications of the Integrals
  • Calculus – Continuity and Differentiability
  • Calculus – Differential Equations
  • Calculus – Integrals
  • Geometry – Area
  • Geometry – Circles
  • Geometry – Conic Sections
  • Geometry – Constructions
  • Geometry – Introduction to Euclid’s Geometry
  • Geometry – Three-dimensional Geometry
  • Geometry – Lines and Angles
  • Geometry – Quadrilaterals
  • Geometry – Straight Lines
  • Geometry – Triangles
  • Linear Programming
  • Matrices and Determinants
  • Mensuration – Areas
  • Mensuration – Surface Areas and Volumes
  • Number Systems
  • Number Systems – Real Numbers
  • Permutations and Combinations
  • Probability
  • Sequence and Series
  • Sets and Functions
  • Statistics 
  • Trignometry – Height and Distance
  • Trignometry – Identities
  • Trignometry – Introduction

Middle School Science

  • Acids, Bases And Salts
  • Air and Its Constituents
  • Basic Biology
  • Body Movements
  • Carbon and Its Compounds
  • Cell – Structure And Functions
  • Changes Around Us
  • Chemical Effects Of Electric Current
  • Chemistry in Your Life
  • Coal And Petroleum
  • Combustion And Flame
  • Components Of Food
  • Conservation Of Plants And Animals
  • Crop Production And Management
  • Electric Current And Its Effects
  • Electricity And Circuits
  • Elements and Compounds
  • Fibre To Fabric
  • Food production and management
  • Force And Pressure
  • Forests: Our Lifeline
  • Friction
  • Fun With Magnets
  • Garbage In, Garbage Out
  • Getting To Know Plants
  • Health and Hygiene
  • Heat
  • Hydrogen
  • Life Processes: Nutrition in Animals and Plants
  • Light, Shadows And Reflections
  • Materials: Metals And Non-Metals
  • Matter and Its States
  • Metals and Non-metals
  • Micro Organisms: Friend And Foe
  • Motion And Measurement Of Distances
  • Motion And Time
  • Nutrition In Animals
  • Nutrition In Plants
  • Organization in Living Things
  • Our Environment
  • Physical And Chemical Changes
  • Pollution and conservation
  • Pollution Of Air And Water
  • Reaching The Age Of Adolescence
  • Reproduction In Animals
  • Reproduction In Plants
  • Respiration In Organisms
  • Rocks and Minerals
  • Separation Of Substances
  • Simple Machines
  • Soil
  • Some Natural Phenomena
  • Sorting Materials Into Groups
  • Sound
  • Stars And The Solar System
  • Structure of Atom
  • Synthetic Fibers And Plastics
  • The Living Organisms And Their Surroundings
  • Transfer of Heat
  • Transformation of Substances
  • Transportation In Animals And Plants
  • Universe
  • Waste-water Story
  • Water: A Precious Resource
  • Weather, Climate And Adaptations Of Animals To Climate
  • Winds, Storms And Cyclones

Middle School Math

  • Addition
  • Area and Its Boundary
  • Boxes and Sketches
  • Data Handling
  • Fun With Numbers
  • Heavy and Light
  • How Many
  • Long And Short
  • Mapping
  • Measurement
  • Money
  • Multiplication and Factors
  • Multiply and Divide
  • Numbers
  • Parts and Wholes
  • Pattern Recognition
  • Patterns
  • Play With Patterns
  • Rupees And Paise
  • Shapes And Angles
  • Shapes And Designs
  • Shapes and Space
  • Similarity
  • Smart Charts
  • Squares
  • Subtraction
  • Tables And Shares
  • Tenths and Hundredths
  • Time
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