Coordinate Geometry

Coordinate Geometry

Edited By Team Careers360 | Updated on May 26, 2023 11:46 AM IST

What Is Coordinate Geometry

Coordinate geometry is the branch of mathematics in which the position of a point is described in a plane. It consists of one of the most important parts of mathematics. By the definition of coordinate geometry, you can easily analyse the importance of this chapter wherever be a plane it simply means there is coordinate. It provides a relation between algebra and geometry through graphs of lines and curves. It mainly helps us to locate the points in a plane. Its uses are spread in all fields like trigonometry, calculus, dimensional geometry, etc. The subject has obvious applications in statistics, physics also in real life for the construction field, the sketch of the building is pure geometry, in astrophysics to find the distance between the planets, coordinate geometry helps a lot. In JEE Mains or other entrance examinations, the examiner tries to check your strength by using the concept of coordinate geometry. In physics, various laws and principles, including Newton's Laws of Motion, the Gravitational Law, and the study of forces, utilize concepts from different branches of mathematics, such as plane coordinate geometry, to describe and analyze physical phenomena accurately.

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Also read - NCERT Exemplar Class 10 Maths Solutions Chapter 7 Coordinate Geometry

Why Coordinate Geometry?

Geometry, as a logical system, is a means and even the most powerful means to make children feel the strength of the human spirit that is of their own spirit.- H. FREUDENTHAL

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Coordinate Geometry Real-Life Example

The elements of coordinate geometry" is a branch of geometry that combines algebraic concepts and techniques with geometric principles. In your childhood days, you used to play a Ludo Game In this game, you followed the longest path. Have you ever thought about why you are using this path if your only target is to put the token into the endpoint as we have another option to choose the smallest path which directly puts our token into the endpoint.

The answer is to make the game more interesting, so they defined some coordinates in math in the game where we can save our tokens and it's mandatory to cross all those coordinates to make the game more interesting.

Ohhh! you are studying co ordinate geometry from your childhood

After studying this chapter you definitely understand how Examiner has set Ludo for you.

Now take another coordinate geometry examples

Let's suppose you and your school friends are planning to go for a tour.

The first thing that comes in your mind is to locate a meeting point and the meeting point should be the shortest distance for all.

So how do you decide the meeting point?

Firstly you start sharing their home location.

Secondly, you decide on a location where everyone can easily reach the reason for reaching depends upon the availability of transport.

Thirdly, you choose a point in the direction of your tour so that it consumes less time.

The whole planning is done by using coordinate geometry.

Also Read |Coordinate Geometry Questions For CAT Exam

Coordinate Geometry Notes

Straight Lines: In earlier classes, you are familiar with Straight Lines. Now in this topic, you will study some important properties of the line when straight lines are defined in coordinates.

The inclination of the Line: In a coordinate plane, a line cuts the x-axis and makes two angles with the x-axis, which are supplementary. The angle made by the line with the positive direction of the x-axis and measured anticlockwise is called the inclination of the line.

Circle: A circle is defined as the set of all points in a coordinate plane that is equidistant from a fixed point in the plane.

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Parabola: A parabola is defined as the set of all points in a coordinate plane that is equidistant from a fixed point and a fixed straight line in the plane.

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Ellipse: An ellipse is defined as the set of all points in a plane such that the sum of whose distances from two fixed points in the plane is a constant.

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Hyperbola: A Hyperbola is the set of all points in a plane such that the difference of whose distances from two fixed points in the plane is a constant.

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Coordinate Geometry Important Equations

Equation of a line is
Equation of a circle is , where r is the radius of a circle.
Equation of a parabola is .
Equation of an Ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ .
Equation of a Hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ .

Important Equations of Coordinate Geometry

Distance formula $D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
Slope, m of a line is $m=-\frac{a}{b}$
Angle between two lines is $\theta=tan^{-1}\left |\frac{m_{2}-m_{1}}{1+m_{1} m_{2}} \right |$
Distance, d of a Point (x1,y1) From a Line is $d=\frac{\left|\mathrm{a} x_{1}+\mathrm{b} y_{1}+\mathrm{c}\right|}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}$
Distance between two parallel lines of slope m is $d=\frac{\left|c_{1}-c_{2}\right|}{\sqrt{1+m^{2}}}$

Important Terms of Coordinate Geometry

Coordinate of a point in a plane
Slope and gradient
The angle between two intersecting lines, their intersection point, parallel lines and collinear lines
Focus and Eccentricity
Directrix
Latus Rectum

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Tips That Will Help You In Preparing Coordinate Geometry

The best way to solve the question is by using simple methods.
Always keep in mind the general equation of geometry, if you are unable to do any question then try to make the given equation similar to the general equation for eg. ${y^2}=4(x+y)$ every one get whose equation is this, if you see that degree of y is 2 and the degree of x is 1 so it is the equation of parabola and now try to make it as similar as general equation ${y^2}=4x+4y={y^2}-4y-4=4x-4=(y-2)^2=4(x-1)$ Now it looks simple and you can easily see the value of a is 1.
Important terms defined in this topic is the basic key point of solving all types of problems.
Short trick for solving the multiple choice question is to put the coordinate points from the option to the given question. Remember one thing the given option can trick you at any place so be careful.

Also Read | NCERT Solution for Class 10 Maths Chapter 7 Coordinate Geometry

Best Book for Coordinate Geometry:

Maths NCERT Books are one of the most important study materials as this book covers all the topics. Starting from the NCERT book, the example given in NCERT is simple and lucid. Most of the important concepts and theories you will understand by simply solving those given examples. And also solve all problems (including miscellaneous problems) of NCERT. If you do this, your basic level of preparation will be completed.

Then you can refer to the books Coordinate by Amit M. Agarwal, Cengage Mathematics Coordinate Geometry or RD Sharma. Coordinate Geometry explained very beautifully in the book Arihant Algebra and there are lots of questions with crystal clear concepts. But again the choice of reference book depends on you, find the book that best suits you the best depending on how well you are clear with the concepts and the difficulty of the questions you require. Rather than referring to all the books just stick to one good book but for practising more problems you can refer to other books.

Also, check for | NCERT 2023 – NCERT Solutions, Books, Syllabus, NCERT Exemplar Problems with Solutions

Maths Chapter-wise Notes for Engineering exams

Chapters	Chapters Name
Chapter 1	Sets, Relations, and Functions
Chapter 2	Complex Numbers and Quadratic Equations
Chapter 3	Matrices and Determinants
Chapter 4	Permutations and Combinations
Chapter 5	Binomial Theorem and its Simple Applications
Chapter 6	Sequence and Series
Chapter 7	Limit, Continuity, and Differentiability
Chapter 8	Integral Calculus
Chapter 9	Differential Equations
Chapter 11	Three Dimensional Geometry
Chapter 12	Vector Algebra
Chapter 13	Statistics and Probability
Chapter 14	Trigonometry
Chapter 15	Mathematical Reasoning
Chapter 16	Mathematical Induction

Topics from Co-ordinate geometry

Cartesian system of rectangular coordinates in a plane ( JEE Main, KCET, 3+ More) (16 concepts)

Distance formula, section formula, locus and its equation, translation of axes ( JEE Main, KCET, 3+ More ) (96 concepts)

Intersection of lines, slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinates axes ( JEE Main, KCET, 3+ More ) (32 concepts)

Various forms of equations of a line ( JEE Main, KCET, 3+ More ) (80 concepts)

Angles between two lines, conditions for concurrence of three lines, distance of a point from a line ( JEE Main, KCET, 3+ More ) (32 concepts)

Equations of internal and external bisectors of angles between two lines ( JEE Main, KCET, 3+ More ) (12 concepts)

Equation of family of lines passing through the point of intersection of two lines ( JEE Main, KCET, 3+ More ) (12 concepts)

Standard form of equation of a circle, general form of the equation of a circle, its radius and centre ( JEE Main, KCET, 3+ More ) (88 concepts)

Equation of a circle when the endpoints of a diameter are given ( JEE Main, KCET, 3+ More ) (24 concepts)

Point of intersection of a line and a circle with the centre at the origin and condition for a line to be tangent to a circle ( JEE Main, KCET, 3+ More ) (20 concepts)

Equation of the tangent ( JEE Main, KCET, 3+ More ) (24 concepts)

Sections of cones, equations of conic sections(parabola, ellipse and hyperbola) in standard forms ( JEE Main, KCET, 3+ More ) (141 concepts)

Conditions for Y=mx+c to be a tangent and point(s) of tangency ( JEE Main, KCET, 3+ More ) (28 concepts)

Coordinates of a point in space, distance between two points ( AEEE, JEE Main, 2+ More ) (30 concepts)

Sections formula, direction ratios and direction cosines, angles between two intersecting lines ( AEEE, JEE Main, 2+ More ) (39 concepts)

Skew lines, the shortest distance between them and its equation ( AEEE, JEE Main, 2+ More ) (27 concepts)

Equations of a line and plane in different forms, intersection of a line and a plane, coplanar lines ( AEEE, JEE Main, 2+ More ) (123 concepts)

Vectors and scalars, addition of vectors, components of a vector in two dimensions and three dimensional space ( AEEE, JEE Main, 5+ More ) (34 concepts)

Scalar and vector products, scalar and vector triple product ( AEEE, JEE Main, 5+ More ) (70 concepts)

Coordinate System ( JEE Main, KCET, 3+ More ) (44 concepts)

Straight Line ( JEE Main, KCET, 3+ More ) (84 concepts)

Circle ( JEE Main, KCET, 3+ More ) (92 concepts)

Parabola ( JEE Main, KCET, 3+ More ) (100 concepts)

Ellipse ( JEE Main, KCET, 3+ More ) (60 concepts)

Hyperbola ( JEE Main, KCET, 3+ More ) (70 concepts)

Line ( AEEE, JEE Main, 2+ More ) (6 concepts)

Plane ( AEEE, JEE Main, 2+ More ) (16 concepts)

Line and Plane ( AEEE, JEE Main, 2+ More ) (6 concepts)

Sphere ( AEEE, JEE Main, 2+ More ) (2 concepts)

Frequently Asked Questions (FAQs)

1. What is coordinate geometry meaning?

Coordinate geometry is a branch of mathematics that deals with the study of geometric figures using a coordinate system. It involves representing points, lines, curves, and shapes on a plane using numerical coordinates.

2. How does the coordinate system work in coordinate geometry?

The coordinate system in coordinate geometry consists of an x-axis and a y-axis that intersect at a point called the origin (0,0). Points on the plane are identified by their coordinates, which are written as (x, y), where x represents the horizontal distance from the y-axis (positive to the right and negative to the left) and y represents the vertical distance from the x-axis (positive upward and negative downward).

3. What are the formulas for distance and midpoint in coordinate geometry?

The distance formula between two points (x₁, y₁) and (x₂, y₂) in coordinate geometry is given by:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

The midpoint formula between two points (x₁, y₁) and (x₂, y₂) is given by:

((x₁ + x₂)/2, (y₁ + y₂)/2)

4. How do you find the slope of a line in coordinate geometry?

The slope of a line in coordinate geometry is a measure of its steepness and is given by the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Here, (x₁, y₁) and (x₂, y₂) are two points on the line.

5. What is the equation of a line in coordinate geometry?

The equation of a line in coordinate geometry can be written in different forms, including the slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept (the point where the line intersects the y-axis). Other forms include the point-slope form and the general form. The equation of a line can be determined using various information, such as the slope and a point on the line or two points on the line.

6. Name some coordinate geometry project?

Here are some potential project ideas in coordinate geometry:

Graphing Equations: Create visual representations of various equations using coordinate geometry, exploring different types of functions and their graphs.
Transformations: Investigate geometric transformations such as translations, rotations, reflections, and dilations in coordinate geometry, analyzing how they affect the positions of points and shapes.
Real-Life Applications: Explore practical applications of coordinate geometry, such as using GPS systems, satellite tracking, or analyzing motion in physics, to understand how coordinates play a crucial role in real-world scenarios.