COORDINATE GEOMETRY
Straight Lines
The equations of straight lines come in two forms:
1. y = mx + c, where m is the gradient and c is the
y-intercept.
2. ax + by + c = 0, where a, c and c are integers.
COORDINATE GEOMETRY
Straight Lines
When equations are in the form ax + by + c = 0 they
can be rearranged into the form y = mx + c so that
the gradient and the y-intercept can be found
easily.
Example:
Write 2x + 3y + 5 = 0 in the form y = mx + c and
state the gradient and y-intercept of the line.
COORDINATE GEOMETRY
Straight Lines
Points of Revision:
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.
When two lines are parallel they have the same
gradient.
COORDINATE GEOMETRY
Straight Lines
Examples:
Write these lines in the form ax + by + c = 0
(i) y = 2x + 3
(ii) y = 1/4x – 3
COORDINATE GEOMETRY
Straight Lines
Examples:
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.
COORDINATE GEOMETRY
Straight Lines
Examples:
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.
COORDINATE GEOMETRY
Straight Lines
Examples:
The line y = 3x – 12 meets the x-axis at the point P.
Find the coordinates of P.
COORDINATE GEOMETRY
Straight Lines
Finding the gradient when given two points
If given two points on a line, (x1, y1) and (x2, y2), we
can find the gradient of the line by using the
formula:
y2 – y1
m = -------x2 – x1
COORDINATE GEOMETRY
Straight Lines
Examples:
Work out the gradient of the line joining the following
points:
(i) (3, 4) and (5, 6)
(ii) (3a, -2a) and (4a, 2a)
COORDINATE GEOMETRY
Straight Lines
Examples:
The line joining (2, -5) to (4, a) has gradient -1.
Work out the value of a.
COORDINATE GEOMETRY
Finding the Equation of a Straight Line
If given the gradient of a line and a point, (x1, y1), on
the line we can find the equation of line using the
formula:
y – y1 = m(x – x1)
COORDINATE GEOMETRY
Straight Lines
Examples:
Find the equation of the line with gradient 4 that
passes through the point (1, 3).
COORDINATE GEOMETRY
Straight Lines
Examples:
Find the equation of the line with gradient -½ that
passes through the point (5, 3).
COORDINATE GEOMETRY
Straight Lines
Examples:
The line y = 4x – 8 meets the x-axis at the point A.
Find the equation of the line with gradient 3 that
passes through the point A.
COORDINATE GEOMETRY
Finding the Equation of a Straight Line
If given the two points on a line, (x1, y1) and (x2, y2),
we can find the equation of line using the formula:
y – y1
x – x1
------- = ------y2 – y1
x2 – x1
COORDINATE GEOMETRY
Straight Lines
Examples:
The find the equation of the line that passes through
the points (1, 2) and (5, 4).
COORDINATE GEOMETRY
Straight Lines
Examples:
The lines y = 4x – 7 and 2x + 3y -21 = 0 intersect at
the point A. The point B has coordinates (-2, 8).
Find the equation of the line that passes through the
points A and B. Write your answer in the form
ax + by + c = 0.
COORDINATE GEOMETRY
Perpendicular Lines
If two lines are perpendicular then
Gradient of line 1 x Gradient of line 2 = -1
Thus, if the gradient of line 1 = m,
then the gradient of line 2 = -1/m
COORDINATE GEOMETRY
Examples:
Work out the gradient of the line that is perpendicular
to the lines with these gradients:
(i) 3
(ii)
-4
(iii)
-½
COORDINATE GEOMETRY
Examples:
Show that the lines y = 2x + 5 and x + 2y + 6 = 0 are
perpendicular.
COORDINATE GEOMETRY
Examples:
Determine whether the lines y – 3x + 3 = 0 and 3y + x
= 6 are parallel, perpendicular or neither.
COORDINATE GEOMETRY
Examples:
Line L is perpendicular to the line 2y – x + 3 = 0 at
the point (4, ½). Determine the equation of the line L.
COORDINATE GEOMETRY
SUMMARY
Equations of lines can be written in the form:
ax + by + c = 0
or
y = mx + c
Given two points we can find the gradient of the line
joining the points using the formula
y2 – y1
m = -------x2 – x1
COORDINATE GEOMETRY
SUMMARY
Given a point on a line and its gradient we can find the
equation of the line using the formula:
y – y1 = m(x – x1)
Given two points on a line we can find the equation of
the line using the formula:
y – y1
x – x1
------- = ------y2 – y1
x2 – x1