Today we are here to discuss Coordinate Geometry from the root of it. So, The whole article is about what Coordinate Geometry is, relevant problems and their solutions as much as possible.
Table of Contents:
Coordinate Geometry is the most interesting and important field of Mathematics. It is used in physics, engineering and also in aviation, rocketry, space science, spaceflight etc.
To know about Coordinate Geometry first we have to know what Geometry is.
In greek ‘Geo’ means Earth and ‘Metron’ means Measurement i.e. Earth Measurement. It is the most ancient part of mathematics, concerned with the properties of space and figures i.e positions, sizes, shapes, angles and dimensions of things.
What is Coordinate Geometry?
Coordinate geometry is the way of learning of geometry using the Co-ordinate system. It describes the relationship between geometry and algebra.
Many mathematicians also called Coordinate geometry as Analytical Geometry or Cartesian Geometry.
Why is it called Analytical Geometry?
Geometry and Algebra are two different branches in Mathematics. Geometrical shapes can be analyzed by using algebraic symbolism and methods and vice versa i.e. algebraic equations can be represented by Geometric graphs. That is why it is also called Analytical Geometry.
Why is it called Cartesian Geometry?
Coordinate Geometry was also named Cartesian Geometry after French mathematician Rene Descartes as he independently invented the cartesian coordinate in the 17th century and using this, put Algebra and Geometry together. For such a great work Rene Descartes is known as the Father of Coordinate Geometry.
(B) Coordinate system
A Coordinate system is the base of Analytical Geometry. It is used in both two dimensional and three-dimensional fields. There are four types of coordinate system in general.
C. The whole subject of coordinate Geometry is divided into two chapters.
- One is ‘Coordinate Geometry in Two Dimensions’.
- The second one is ‘Coordinate Geometry in Three Dimensions’.
Coordinate Geometry in Two Dimensions (2D):
- Here we are going to discuss both the Cartesian and Polar Coordinates in two dimensions one by one. We will also solve some problems to get a clear idea of the same, and later we will find the relation between them as well.
Cartesian Coordinate in 2D:
At first, we will have to learn the following terms through graphs.
i) Coordinate Axes
Read and Follow the Figures simultaneously.
Suppose the horizontal line XX
and vertical line YY are two perpendicular lines intersecting each other at right angles at the point O , XX
and YY are number lines, the intersection of XX
and YY forms XY-plane and P is any point on this XY-plane.
Coordinate Axes in 2D
and YY are described as the Coordinate Axes. XX
is indicated by X-Axis and YY is indicated by Y-Axis. Since XX
and YY are number lines, the distances measured along OX and OY are taken as positive and also the distances measured along OX
and OY are taken as negative. (See above graph.1)
What is Origin in 2D?
The point O is called the Origin. O is always supposed to be the starting point. To find the position of any point on the coordinate plane we always have to begin the journey from the origin. So the origin is called the Zero Point. (Please refer the above graph.1)
What do we understand by a Coordinate Plane?
The XY plane defined by two number lines XX
and YY or the X-axis and Y- axis is called the Coordinate Plane or Cartesian Plane. This Plane extends infinitely in all direction. This is also known as two-dimensional plane. (See above graph.1)
*Assume the variables x>0 and y>0 in the above figure.
What is Coordinate in 2D?
Coordinate is a pair of numbers or letters by which the position of a point on the coordinate plane is located. Here P is any point on the coordinate plane XY. The coordinates of the point P is symbolized by P(x,y) where x is the distance of P from Y axis along X axis and y is the perpendicular distance of P from X axis respectively. Here x is called the abscissa or x-coordinate and y is called the ordinate or y-coordinate (See above Graph 2)
How to Plot a Point on the coordinate plane?
Always we will have to start from the origin and first walk towards right or left along X axis to cover the distance of x-coordinate or abscissa ,then turn the direction up or down perpendicularly to the X axis to cover the distance of ordinate using units and their signs accordingly. Then we reach the required point .
Here to represent the given point P(x,y) graphically or to plot it on the given XY plane, first start from the origin O and cover the distance x units along X axis (along OX) and then turn at 90 degree angle with X axis or parallelly to Y axis(here OY) and cover the distance y units . (See above graph 3)
How to find coordinates of a given point in 2D ?
Let XY be the given plane,O be the origin and P be the given point.
First draw a perpendicular from the point P on X axis at the point A. Suppose OA=x units and AP=y units, then the Coordinates of the point P becomes (OA , AP) i.e. (x,y).
Similarly if we draw another perpendicular from the point P on Y axis at the point B, then BP=x and OB=y.
Now since A is the point on the X axis ,the distance of A from Y axis along X axis is OA=x and perpendicular distance from X axis is zero,so the coordinates of A becomes (x,0).
Similarly, the coordinates of the point B on the Y axis as (0,y) and the coordinates of Origin O is (0,0).
Graph 5 * colour green denotes the beginning
What is Quadrant in 2D?
Coordinate Plane is divided into four equal sections by the coordinate axes. Each section is called Quadrant. Going around counterclockwise or anticlockwise from upper right, the sections are named in the order as Quadrant I, Quadrant II, Quadrant III and Quadrant iv.
Here we can see the X and Y axes divide the XY plane into four sections XOY, YOX
and YOX accordingly. Therefore, the area XOY is the Quadrant I or first quadrant, YOX
is the Quadrant II or second quadrant, XOY
is the Quadrant III or third quadrant and YOX is the Quadrant IV or fourth quadrant.(please refer the graph 5)
Points in Different Quadrants of coordinate plane:
Since OX is +ve and OX
is -ve side of X axis and OY is +ve and OY is -ve side of Y axis, signs of coordinates of points in different quadrants—-
Quadrant I: (+,+)
Quadrant II: (-,+)
Quadrant III: (-,-)
Quadrant IV: (+,-)
For example, if we go along OX from O and draw a perpendicular from any point P in the Quadrant I on the X axis (OX) at the point A so that OA=x and AP=y then coordinate of P is defined as (x,y) as described in the article (How to find coordinate of a given point?).
Again if we go along OX
from O and draw a perpendicular from any point Q in the Quadrant II on the X axis (on OX) at the point C so that OC=x and CQ=y then the coordinates of Q is defined as (-x,y).
Similarly the coordinates of any point R in quadrant III is defined as (-x,-y) and the coordinates of any point in quadrant IV is defined as (x,-y). (see graph 6)
The brief information about Coordinate Geometry with basic concepts has been provided to get a clear idea to start the subject. We will subsequently discuss details about 2D and 3D in the upcoming posts. If you want further study go through:
- 1. https://en.wikipedia.org/wiki/Analytic_geometry
- 2. https://en.wikipedia.org/wiki/Geometry
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