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Coordinate
geometry Cont'd
Clement
Radcliffe, Contributor
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| Members
of the Calabar High Schools' Challenge
Quiz team. From left are Kemoy
Lindsay, Demar Barnett (captain),
Rexander Wright and Jermaine Robinson.
- Ricardo Makyn Photo |
LAST
WEEK, we began the review of the Cartesian
plane by considering gradient, midpoint,
intercept and finding the length of
the line. The solution of the homework
will further illustrate this.
Find
the gradient, midpoint and length
of the line joining the points:
A(-8,2),
B(3,-2).
SOLUTION
Gradient
Midpoint
Length
of line
We
will continue coordinate geometry
by considering:
EQUATION
OF STRAIGHT LINES
Reminders
All straight lines have the equation:
y = mx + c where m is the gradient
and c is the intercept.
When the axes cut at the origin (0,
0), the equation of the x axis is
y = 0 and for the y axis it is x =
0.
y= 2x + 3 is the equation of a line
if for each point (x, y) on the line,
the y coordinate is equal to twice
the x coordinate of the same point
plus 3. The points (2, 7) and (-1,
1) are, therefore, on the line.
This
fact about an equation is not usually
emphasised but must be clearly noted.
The point (x, y) is on the line y
= mx + c if it satisfies the equation.
You
may show that (1, -2) is a point on
the line y = 3x - 5 by substituting
x = 1 and y = -2 into the equation.
(Substitution shows that - 2 = -2).
The value of c, the intercept of a
line, is found by substituting x =
0 into its equation. Do you know why?
METHODS
OF FINDING THE EQUATION
The
following are the three methods which
are commonly used to find the equation
of a straight line.
(i)
Evaluating the equation given the
gradient m and the intercept c.
Example
Find
the equation given that m =3/4 and
c = 1.
The
equation is y = mx + c. Substituting
y = 3x/4 + 1 or 4y = 3x + 4
Answer
is 4y = 3x + 4.
This
method can be extended to a given
line on a graph. In this case, both
the gradient and the intercept can
be found from the graph and the equation
determined.
(ii)
Given the coordinates of two points,
(x1,y1)
and (x2, y2),
the equation is
Using
the points A(4, -1), B(1 ,1) in the
above, then
(iii)
The formula given in (ii) may be expressed
as y- y1 = m
x-x1
I
am sure you realise that m = y2 -y1
x2-x1
This
formula is used, given the coordinates
of a point on the line and the gradient
of the line.
Example
Find
the equation of the line if the gradient
m = 2/3 and the point (1, 2) is on
the line.
Now
for your homework.
1.
A straight line HK cuts the y axis
at H(0, -1). The gradient of HK is
2/3.
Show
that the equation of the line HK is
2x - 3y = 3.
No.
7(b) CXC, June 1994 General Proficiency
2.
A straight line is drawn through the
points A(-5, 3) and B(1, 2).
(i)
Determine the gradient of AB.
(ii)
Write the equation of the line AB.
(No. 4(a) CXC, June 1995 General)
I
must emphasise again that the problems
based on this topic are fairly routine.
It will do you well to practise them
so as not to miss out on the opportunity
to score full marks for the question
if it is presented in the June exam
this year.
*
Clement Radcliffe is principal
of Glenmuir High School in Clarendon.
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