Co-ordinate geometry  

What is Co-ordinate 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 analyze 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. And 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, Newton's Law, of Motion, Gravitational's Law, All types of forces even in measurement Coordinate geometry is used.

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

 

Co-ordinate Geometry real-life example

In your childhood days, you used to play a Ludo Game

In this game, you followed the longest path. Have you ever think 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 put our token into the endpoint.

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

Ohhh! you are studying coordinate geometry from your childhood 

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

Now take another example

Let 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 shortest distance fo all.

So how 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.

 

Notes of Coordinate Geometry 

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 x-axis and measured anti-clockwise 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.

 

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.

 

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.

 

 

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. 

 

Important Equations of Coordinate Geometry

  • Equation of a line is ax+by+c=0
  • Equation of a circle is  x^2+y^2=r^2, where r is the radius of a circle.
  • Equation of a parabole is  y^2=4ax.
  • 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 ax+by+c=0 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 ax+by+c=0 isd=\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

 

Tips that will help you in preparing Coordinate Geometry

  • The best way is 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 its look 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 that put the coordinate points from the option to the given question remember one thing the given option can tricked you at any place so be careful.

 

Best Book for Coordinate Geometry :

Maths NCERT Books are one of the most important study material as this book covers all the topics. Start from NCERT book, the example given in NCERT is simple and lucid. Most of the important concepts and theory you will understand by simply solving those given example. And also solve all problems (including miscellaneous problem) 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 all the books just stick to one good book but for practicing more problem you can refer to other books.

 

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 co-ordinates in a plane ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (16 concepts)
  • Distance formula, section formula, locus and its equation, translation of axes ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (96 concepts)
  • Intersection of lines, slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinates axes ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (32 concepts)
  • Various forms of equations of a line ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (80 concepts)
  • Angles between two lines, conditions for concurrence of three lines, distance of a point from a line ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (32 concepts)
  • Equations of internal and external bisectors of angles between two lines ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (12 concepts)
  • Equation of family of lines passing through the point of intersection of two lines ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (12 concepts)
  • Standard form of equation of a circle, general form of the equation of a circle, its radius and centre ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (88 concepts)
  • Equation of a circle when the end points of a diameter are given ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (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, TS EAMCET, AP EAMCET ) (20 concepts)
  • Equation of the tangent ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (24 concepts)
  • Sections of cones, equations of conic sections(parabola, ellipse and hyperbola) in standard forms ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (141 concepts)
  • Conditions for Y=mx+c to be a tangent and point(s) of tangency ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (28 concepts)
  • Coordinates of a point in space, distance between two points ( AEEE, JEE Main, MHT-CET ) (30 concepts)
  • Sections formula, direction rations and directio cosines, angles between two intersecting lines ( AEEE, JEE Main, MHT-CET ) (39 concepts)
  • Skew lines, the shortest distance between them and its equation ( AEEE, JEE Main, MHT-CET ) (27 concepts)
  • Equations of a line and plane in different forms, intersection of a line and a plane, coplaner lines ( AEEE, JEE Main, MHT-CET ) (123 concepts)
  • Vectors and scalars, addition of vectors, components of a vector in two dimensions and three dimensional space ( AEEE, JEE Main, SRMJEEE, TS EAMCET, VITEEE, AP EAMCET ) (34 concepts)
  • Scalar and vector products, scalar and vector triple product ( AEEE, JEE Main, SRMJEEE, TS EAMCET, VITEEE, AP EAMCET ) (70 concepts)
  • Coordinate System ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (11 concepts)
  • Straight Line ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (21 concepts)
  • Circle ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (23 concepts)
  • Parabola ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (25 concepts)
  • Ellipse ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (15 concepts)
  • Hyperbola ( JEE Main, KCET, TS EAMCET, AP EAMCET ) (25 concepts)

Important Books for Co-ordinate geometry

  • Co-ordinate geometry Book
  • Co-ordinate geometry Book
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