Coordinate Geometry

Have you ever wondered why Earth is marked with imaginary lines i.e, Latitudes and Longitudes that makes up the Earth’s grid system? This is because dividing the earth into grids helps us to locate positions on Earth easily. Global Positioning System (GPS) also uses coordinate geometry to navigate, commute and access the positions without a hitch.


What is Grid?

Grid is a network of horizontal and vertical lines in series that divides the entire space into small blocks. The horizontal rows and vertical columns are either named or numbered. The position of any object on the grid can be specified by the rows and columns it belongs to. For example, in the grid given below, the silver ball belongs to row ‘5’ and column ‘E’. Thus, we can specify the position of the ball either as 5E or E5.


What is Coordinate Geometry?

Coordinate Geometry is a part of geometry that uses two or more numbers to specify the position of any point, figure or object. The position of the object can be defined in a line or a plane or three dimensional space and so on. Below are the types of Coordinate System.


  1. Number Line - It is one of the simplest form of coordinate system where the position of a point can is specified by a number on the given line. Number line is a line that can be extended on either ways such that the origin (middle) of the line is zero, the numbers on the left side are negative and the numbers on the right side of zero are negative. 

  1. Cartesian Coordinate System - It is the coordinate system where the position of the point or object is defined by two or more axis. There are two types of Cartesian Coordinate System:

  1. Plane - In this system the object is not limited to a line. The position of the object anywhere on a plane can be defined by two perpendicular number lines named as x-axis (horizontal line) and y-axis (vertical line). 

  1. Three-Dimensional - In three Dimension, the position of a point in a space can be specified by three perpendicular number lines called X-axis. Y-axis and Z-axis. 


Coordinate Geometry in Plane

We’ll be particularly using the Cartesian coordinate system – a particular type (among many) which represents each point by using a pair of number (coordinates) which are the signed distances of the point from two fixed perpendicular lines, known as the axes.

Here’s what the Cartesian coordinate system looks like:


With reference to the above figure, here are some terminologies, notations and conventions.

  • The horizontal line is referred to as the X axis, and the vertical as Y axis. The axes divide the plane into four quadrants (labelled in red)

  • The point of intersection of the two axes is referred to as the Origin (O)

  • The coordinates of each point are denoted by an ordered pair of numbers (x, y)

  • The x-coordinate of a point is referred to as its abscissa and the y-coordinate as its ordinate

  • The abscissa of a point is its ‘signed’ distance from the Y-axis. By signed, it means that towards the right of the Y-axis, the abscissa is positive, whereas on the left it is negative. (This is a convention).

  • Similarly, the ordinate of a point is its signed distance from the X-axis.

  • Using the above convention, the origin has the coordinates (0, 0) and we can determine the signs of x and y coordinates of a point (indicated in blue) in the four quadrants

Here is another figure to illustrate the above points.


In the figure above, the coordinates of the point P are (3, 1). This means that it is at a distance of 3 units from the X axis (towards the right), and 1 unit from the Y-axis (above it).


The point Q has the coordinates (-2, -2), implying it is at distance of 2 units from both the axes. The negative x-coordinate tells us that it is towards the left of Y-axis. Similarly, the negative y-coordinate means the that the point is below the X-axis. In coordinate geometry, points are placed on the "coordinate plane" as shown below. It has two scales - one running across the plane called the "x axis" and another a right angles to it called the y axis. (These can be thought of as similar to the column and row in the paragraph above.) The point where the axes cross is called the origin and is where both x and y are zero.


On the x-axis, values to the right are positive and those to the left are negative.

On the y-axis, values above the origin are positive and those below are negative.

A point's location on the plane is given by two numbers,the first tells where it is on the x-axis and the second which tells where it is on the y-axis. Together, they define a single, unique position on the plane. So in the diagram above, the point A has an x value of 20 and a y value of 15. These are the coordinates of the point A, sometimes referred to as its "rectangular coordinates". Note that the order is important; the x coordinate is always the first one of the pair.


For a more in-depth explanation of the coordinate plane see The Coordinate Plane.

For more on the coordinates of a point see Coordinates of a Point


Important Formulas

The Distance Between two Points

Draw a line between the two points. Complete a right angle triangle and use Pythagoras' theorem to work out the length of the line.


Between points A and B:

AB2 = (Bx – Ax)2 + (By – Ay)2

 

The Midpoint of a Line Joining Two Points

The midpoint of the line joining the points (x1, y1) and (x2, y2) is:

  • [½(x1 + x2), ½(y1 + y2)]


Example

Find the coordinates of the midpoint of the line joining (1, 2) and (3, 1).

Midpoint = [½(3 + 1), ½(2 + 1)] = (2, 1.5)

 

The Gradient of a Line Joining Two Points

The gradient of a line joining points (x1, y1) and (x2, y2) is (y2 - y1)/(x2 - x1).

 

Parallel and Perpendicular Lines

If two lines are parallel, then they have the same gradient.

If two lines are perpendicular, then the product of the gradients of the two lines is -1.


Example

a) y = 2x + 1

b) y = -½ x + 2

c) ½y = x - 3

The gradients of the lines are 2, -½ and 2 respectively. Therefore (a) and (b) and perpendicular, (b) and (c) are perpendicular and (a) and (c) are parallel.

 

The Equation of a Line Using One Point and the Gradient

The equation of a line which has gradient m and which passes through the point (x1, y1) is:

  • y - y1 = m(x - x1)


Example

Find the equation of the line with gradient 2 passing through (1, 4).

y - 4 = 2(x - 1)

y - 4 = 2x - 2

y = 2x + 2

Since m = \[\frac{y2-y1}{x2-x1}\]


The equation of a line passing through (x1, y1) and (x2, y2) can be written as:

\[\frac{y-y1}{x-x1}\] = \[\frac{y2-y1}{x2-x1}\]