MATHEMATICS FORM ONE TOPIC 4- COORDINATE GEOMETRY(1)
The position of points on a line found by using a number line, that is
When two number lines one vertical and another one horizontal are considered one kept at 90o and intersecting at their zero marks, The result is called xy – plane or Cartesian plane. The horizontal one is called x – axis and the vertical is called y – axis.
Origin is the where the two axes that is x – axis and y axis (intersect)
Coordinates of a point
The position of a point in the xy – plane is given by a pair of in the form of ordered pair. Thus ordered pair is called coordinate. The coordinate of the point is therefore written in the form of (a, b), Where the first number ‘’a’’ is the value in the horizontal axis i.e x – axis b is the value in the y – axis
The value in the x – axis is also referred to as abscissa and y – axis is called ordinate. All distance in the xy – plane are measured the origin.
Examples write the coordinates of the following point A, B, C, D, E, F
The coordinates of the points are
A = (0,5)
B = (5,0)
C = (0,4)
D = (-5,5)
1. a) Write down the coordinates of each of the labeled points in figure 9.2
b) State the quadrant in which each of these points F, H, V and I belong
2. Draw axes on a graph paper and plot the points given below. Join in the order given with straight lines forming polygonal figures shape have you drawn in each case.
We have already discussed how to find the gradient of a line for example the gradient of the line joining points (2, – 4) and (5,0) is given as.
Since the two points are collinear.
we can find the equation of the line having any point on a line say (x,y) and any point,
let (x1,y1) = (x,y), and (x2,y2) = (5,0)
4(5-x) = 3x-y
20-4x = -3y
In general the equation of a straight line is written as y = mx + c. Where m – Is the slope of the line and c is ordinate of the y. called y- intercept
The point on the line (x,y) is called arbitrary point
Example: – find the equation of line passing through the points.
(12,-6) and (2, 6)
(2) Give that y = –
+ 6 find the gradient of this line
The gradient is
Example: find the equation of the line passing through the point (4, 6) and having a slope -1/2
(x,y) , (4,6) , M =
2(y-6) = x – 4
2y – 12 = x – 4
2y = x + 8
X-intercept and y – intercept.
X-intercept is the point where a line meets (cuts) the x-axis, at the value of y (ordinate) is equal to zero.
That is to say the x-intercept is found by substituting y = 0 in the equation. Therefore for the equation y = mx + c.
Y = 0, 0 = mx + c
Mx + c = 0
Mx = -c
Therefore the coordinate of x-intercept is (-c/m, 0).
y- Intercept is the point where the line and the y- axis meet. All this point the abscissa is normally equal to zero. The x- intercept is found by setting. x=0
i.e y = mx + c
y = m(0) + c
y = c
The intercept (0,c)
The coordinate of the Y-intercept is (0,c)
Example : – a line L is passing through the points A(5 – 2) and B(1,4).
i. The equation of the line in the form of
Y= mx +c and ax + by + c = o
ii. The x and y intercept.
(x,y) = (1,4)
-3 + 3x = 8 – 2y
2y = 11- 3x
then -3 + 3x = 8 – 2y
-3 + 3x = 8 – 2y
-3 + 3x – 8 + 2y = 0
3x – 11 + 2y = 0
3x + 2y – 11 = 0
(ii) Y – intercept
Let x = 0
3x + 2y – 11 = 0
Y – Intercept =0,
The coordinate of Y-intercept is (0,11/2)
∴ find the x-intercept
mx + c = y
The coordinate of X-intercept is (22/3,0)
(iii) If ax + by = 12 goes through points (1,-2) and (4, 1) find the value a and b-
let the two collinear point be (x,y),(4,1) and gradient 1
x-4 = y-1
if the equation is multiple by 4 both side we have
compare the equations.
x-y=3 and ax+by=12
The value of a=1 and b=-4
3. Find the equations of lines through points
a) (2,1) with gradient 2.
b) (0,5) with gradient -2
c) (1,-3) with gradient-3
d) (-2, -4) with gradient
e) (0, 0 ) with gradient -3
f) (-3 , -3) and y- intercept
g) ( 6, 2) and y intercept -2
h) (-1 , -1 ) and y – intercept –
1. The sum of two number is 109 and the difference of the same numbers 29. find the numbers
2. Two number are such that the first number plus three times the second number is 1. And the first minus three times the second is 1/7. Find the two numbers
3. The sum of the number of boys and girls in a class is 36. If twice the number of girls exceeds the number of boys by 12, find the number of girls and that of boys in the class.
4. Twice the length of a rectangle exceeds three times the width of the rectangle by one centimeter and if one – third of the difference of the length and the width is one centimeter find the dimensions of the rectangle.
5. The cost of 4 pencils and five pens together is 6000 shillings while the cost of 6 pencils and 8 pens is 940 shillings, calculate the cost of one pencil and one pe
6. Half of Paul’s money plus one – fifth of John’s money is 1400 shilling John’s money is 2650 shillings. How much has each?
7. A farmer buys 3 sheep and 4 goats for shs 290,Another buys sheep and goats from the some market for shs 170.What price did they pay for (a) 1 goat (b) 1 sheep