COORDINATE GEOMETRY

In this chapter we will be learning to:

o Find the equation of a line from geometrical

information

o Find the general equation of a line using

points and gradients

o Determine if two lines are parallel or

perpendicular

COORDINATE GEOMETRY

PRE-KNOWLEDGE

Before starting this chapter you should be able to:

oRecognise a linear equation

oPlot a straight line graph

oIdentify the gradient and the y-intercept from the equation of a straight line.

oFind the points of intersection of a line and the co-ordinate axes.

COORDINATE GEOMETRYWeek Commencing Monday 26th October

Learning Intention:

1. To be able to find the equation of a line given the gradient and the y-intercept

2. To be able to find the gradient of a line given two points on the line.

Contents:

1. Equations of Straight Lines

2.Finding the Gradient of a Line Given 2 Points on the Line

3. Assignment 1

4. Pre-Knowledge Revision

COORDINATE GEOMETRY

STRAIGHT LINES

The equation of a straight line generally takes

two forms:

(i) ax + by + c = 0 where a, b and c are

integers

or

(ii) y = mx + c where m is the gradient

and c is the y-intercept

COORDINATE GEOMETRY

STRAIGHT LINES

If an equation is written in the form ax + by + c = 0

it can be rearranged into the form y = mx + c so that

the gradient and the y-intercept can be easily read.

Example:Write 2x + 3y + 5 = 0 in the form y = mx + c and state the gradient and y-intercept of the line.

Solution:Taking the x term and the number to the RHS gives:

by dividing across by 3

Gradient = -2/3 and y-intercept = -5/3

35

x32

y

52x3y

COORDINATE GEOMETRY

STRAIGHT LINES

If we know the gradient of a line and its y-intercept

we can write the equation of the line.

Example:A line is parallel to the line y = 1/3x – 4 and crosses the y-axis at the point (0, 6). Write down the equation of the line.

Solution:As the line is parallel to y = 1/3x – 4 the gradient we need is 1/3.

As the point (0, 6) is on the y-axis the y-intercept = 6.

So, equation of the required line is: y = 1/3x + 6

COORDINATE GEOMETRY

STRAIGHT LINES

Example:A line is parallel to the line 3x + 5y + 1 = 0 and it passes through the point (0, 4). Work out the equation of the line.

Solution:To find the equation of the line we need the gradient and the y-intercept. We have the y-intercept from the point: 4.

To find the gradient we need to re-arrange the equation of the given line. This gives:5y = -3x – 1y = -3/5x – 1/5 therefore required gradient = -3/5

Equation of required line is: y = -3/5x + 4

COORDINATE GEOMETRY

FINDING THE GRADIENT WHEN GIVEN 2 POINTS

If given two points on a line, (x1, y1) and (x2, y2), we can find the gradient of the line using the

formula:

12

12

xxyy

m

COORDINATE GEOMETRY

FINDING THE GRADIENT WHEN GIVEN 2 POINTS

Example:Work out the gradient of the line joining the points

(3, 4) and (5, 6).

Solution:We can use the formula

(3, 4) gives x1 = 3 y1 = 4(5, 6) gives x2 = 5 y2 = 6

Substituting into the formula we get:

12

12

xxyy

m

122

m

3546

m

COORDINATE GEOMETRY

FINDING THE GRADIENT WHEN GIVEN 2 POINTS

Example:The line joining (2, -5) to (4, a) has gradient -1.

Work out the value of a.

Solution:Substituting into the formula for the gradient we

get:

7- 52a

2 by across gmultiplyin 25a

- - 12

5a

1245a

COORDINATE GEOMETRY

Assignment 1

Follow the link for Assignment 1 in the Moodle Course Area underneath Coordinate Geometry. This is a Yacapaca Activity.

Assignment must be completed by 5:00pm on Wednesday 4th November 2009

COORDINATE GEOMETRY

PRE-KNOWLEDGE REVISION

COORDINATE GEOMETRY

Linear Equations

A linear equation is an equation that DOES NOT contain any powers higher than 1.

Example:3x + 2y = 1 is a linear equation

3x2 + 2y = 1 is NOT a linear equation as it contains x2

COORDINATE GEOMETRY

Plotting a Straight Line Graph

To plot a straight line graph:

1. Make a table of values for at least three values of x.

2. Find the corresponding values of y

3. Plot the points found

4. Join points with a straight line

COORDINATE GEOMETRY

Identifying Gradient and y-intercept

If the equation of a line is given as y = mx + c,

the coefficient of x is the gradient

c is the y-intercept (the point where it crosses the y-

axis)

COORDINATE GEOMETRY

Points of Intersection of Lines and Coordinate Axes

To find where a line cuts the x-axis let the equation equal 0.

To find where a line cuts the y-axis let x equal 0 in the equation.