TERMINOLOGY 7 Linear Functions Collinear points: Two or more points that lie on the same straight line Concurrent lines: Two or more lines that intersect at a single point Gradient: The slope of a line measured by comparing the vertical rise over the horizontal run. The symbol for gradient is m Interval: A section of a straight line including the end points Midpoint: A point lying exactly halfway between two points Perpendicular distance: The shortest distance between a point and a line. The distance will be at right angles to the line
Transcript
TERMINOLOGY
7 Linear Functions
Collinear points: Two or more points that lie on the same straight line
Concurrent lines: Two or more lines that intersect at a single point
Gradient: The slope of a line measured by comparing the vertical rise over the horizontal run. The symbol for gradient is m
Interval: A section of a straight line including the end points
Midpoint: A point lying exactly halfway between two points
Perpendicular distance: The shortest distance between a point and a line. The distance will be at right angles to the line
ch7.indd 390 7/10/09 4:03:50 PM
391Chapter 7 Linear Functions
INTRODUCTION
IN CHAPTER 5, YOU STUDIED functions and their graphs. This chapter looks at the linear function, or straight-line graph, in more detail. Here you will study the gradient and equation of a straight line, the intersection of two or more lines, parallel and perpendicular lines, the midpoint, distance and the perpendicular distance from a point to a line.
DID YOU KNOW?
Pierre de Fermat (1601–65) was a lawyer who dabbled in mathematics. He was a contemporary of Descartes, and showed the relationship between an equation in the form Dx By,= where D and B are constants, and a straight-line graph. Both de Fermat and Descartes only used positive values of x , but de Fermat used the x -axis and y -axis as perpendicular lines as we do today.
De Fermat’s notes Introduction to Loci, Method of Finding Maxima and Minima and Varia opera mathematica were only published after his death. This means that in his lifetime de Fermat was not considered a great mathematician. However, now he is said to have contributed as much as Descartes towards the discovery of coordinate geometry. De Fermat also made a great contribution in his discovery of differential calculus.
Class Assignment
Find as many examples as you can of straight-line graphs in newspapers and magazines.
Distance
The distance between two points (or the length of the interval between two points) is easy to fi nd when the points form a vertical or horizontal line.
ch7.indd 391 6/26/09 4:13:39 AM
392 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
Find the distance between 1. ,1 4-^ h and ,1 2- -^ h Solution
Counting along the y -axis, the distance is 6 units.
2. ,3 2^ h and ,4 2-^ h Solution
Counting along the x -axis, the distance is 7 units.
When the two points are not lined up horizontally or vertically, we use Pythagoras’ theorem to fi nd the distance.
ch7.indd 392 6/26/09 4:13:44 AM
393Chapter 7 Linear Functions
EXAMPLE
Find the distance between points ,3 1-^ h and ,2 5-^ h. Solution
5BC = and 6AC = By Pythagoras’ theorem,
5 625 36
61
7.81
c a b
AB
AB 61
2 2 2
2 2 2
`
Z
= +
= +
= +
=
=
You studied Pythagoras’ theorem in Chapter 4.
DID YOU KNOW?
Pythagoras made many discoveries about music as well as about mathematics. He found that changing the length of a vibrating string causes the tone of the music to change. For example, when a string is halved, the tone is one octave higher.
The distance between two points ,x y1 1_ i and ,x y2 2_ i is given by
d x x y y2 12
2 12= - + -_ _i i
ch7.indd 393 7/10/09 3:07:11 AM
394 Maths In Focus Mathematics Extension 1 Preliminary Course
Proof
Let ,x yA 1 1= _ i and ,x yB 2 2= _ i Length AC x x2 1= - and length BC y y2 1= - By Pythagoras’ theorem
AB AC BC
d x x y y
d x x y y
2 2 2
22 1
22 1
2
2 12
2 12
`
= +
= - + -
= - + -
_ __ _
i ii i
EXAMPLES
1. Find the distance between the points ,1 3^ h and ,3 0-^ h . Solution
Let ,1 3^ h be ,x y1 1_ i and ,3 0-^ h be ,x y2 2_ i
d x x y y
3 1 0 3
4 316 9
255
2 12
2 12
2 2
2 2
= - + -
= - - + -
= - + -
= +
=
=
_ _] ]] ]
i ig g
g g
So the distance is 5 units.
2. Find the exact length of AB given that ,A 2 4- -= ^ h and ,B 1 5-= ^ h . Solution
Let ,2 4- -^ h be ,x y1 1_ i and ,1 5-^ h be ,x y2 2_ i
4
d x x y y
1 2 5
1 91 81
82
2
2 12
2 12
2
2 2
= - + -
= - - - + - -
= +
= +
=
_ _^ ^i ih h6 6@ @
If points A and B were changed around, the formula would be
( ) ( ) ,d x x y y1 22
1 22= - + -
which would give the same answer.
You would still get 82 if you used )( 2, 4- - as ( , )x y2 2 and ( ),1 5- as ( , )x y1 1 .
ch7.indd 394 6/26/09 4:13:47 AM
395Chapter 7 Linear Functions
7.1 Exercises
1. Find the distance between points (a) ,0 2^ h and 3,6^ h (b) ,2 3-^ h and ,4 5-^ h (c) ,2 5-^ h and ,3 7-^ h
2. Find the exact length of the interval between points
(a) 2, 3^ h and ,1 1-^ h (b) ,5 1-^ h and 3, 0^ h (c) ,2 3- -^ h and 4,6-^ h (d) ,1 3-^ h and ,7 7-^ h
3. Find the distance, correct to 2 decimal places, between points
(a) ,1 4-^ h and 5,5^ h (b) 0, 4^ h and ,3 2-^ h (c) ,8 1-^ h and ,7 6-^ h
4. Find the perimeter of ABCD with vertices , ,,A B3 1 1 1-^ ^h h and , .C 1 2- -^ h
5. Prove that the triangle with vertices 3, 4^ h , ,2 7-^ h and ,6 1-^ h is isosceles.
6. Show that ,AB BC= where , ,,A B2 5 4 2= - -=^ ^h h and , .C 3 8= - -^ h
7. Show that points ,3 4-^ h and 8,1^ h are equidistant from point , .7 3-^ h
8. A circle with centre at the origin O passes through the point , .2 7_ i Find the radius of the circle, and hence its equation.
9. Prove that the points , ,,X Y2 3 1 10- -_ _i i and ,Z 6 5-_ i all lie on a circle with centre at the origin. Find its equation.
10. If the distance between ,a 1-^ h and 3, 4^ h is 5, fi nd the value of a .
11. If the distance between ,3 2-^ h and 4, a^ h is 7, fi nd the exact value of a .
12. Prove that , , ,A B1 4 1 2^ ^h h and ,C 1 3 3+_ i are the vertices of an equilateral triangle.
13. If the distance between , 3a^ h and 4, 2^ h is 37, fi nd the values of a .
14. The points , , ( , ),M N1 2 3 0- -^ h ,P 4 6^ h and ,Q 0 4^ h form a quadrilateral. Prove that MQ NP= and .QP MN= What type of quadrilateral is MNPQ?
15. Show that the diagonals of a square with vertices , , , , ,A B C2 4 5 4 5 3- -^ ^ ^h h h and ,D 2 3- -^ h are equal.
16. (a) Show that the triangle with vertices , , ,A B0 6 2 0^ ^h h and ,C 2 0-^ h is isosceles.
(b) Show that perpendicular ,OA where O is the origin, bisects BC .
17. Find the exact length of the diameter of a circle with centre ,3 4-^ h if the circle passes through the point ,7 5^ h .
18. Find the exact length of the radius of the circle with centre (1, 3) if the circle passes through the point ,5 2- -^ h .
19. Show that the triangle with vertices , , ,A B2 1 3 3-^ ^h h and ,C 7 7-^ h is right angled .
20. Show that the points , , ,X Y3 3 7 4-^ ^h h and ,Z 4 1-^ h form the vertices of an isosceles right-angled triangle .
ch7.indd 395 6/26/09 4:13:48 AM
396 Maths In Focus Mathematics Extension 1 Preliminary Course
Midpoint
The midpoint is the point halfway between two other points.
The midpoint of two points ,x y1 1_ i and ,x y2 2_ i is given by
,x x y y
M2 2
1 2 1 2=
+ +e o
Proof
Find the midpoint of points ,x yA 1 1_ i and ,x yB 2 2_ i . Let ,x yM = ^ h Then ABR;DAPQ <D
ARAQ
ABAP
` =
`
`
x xx x
x x x xx x x x
x x x
xx x
yy y
21
22 2
2
2
2Similarly,
2 1
1
1 2 1
1 2 1
1 2
1 2
1 2
-
-=
- = -
- = -
= +
=+
=+
_ i
Can you see why these triangles are similar?
EXAMPLES
1. Find the midpoint of ,1 4-^ h and 5, 2^ h . Solution
xx x
21 2
=+
ch7.indd 396 7/10/09 3:07:12 AM
397Chapter 7 Linear Functions
So ( , ) .
yy y
M
21 5
24
2
2
24 2
26
3
2 3
1 2
=- +
=
=
=+
=+
=
=
=
2. Find the values of a and b if ,2 3-^ h is the midpoint between ,7 8- -^ h and ,a b^ h . Solution
So and .
xx x
a
a
a
yy y
b
b
b
a b
2
22
7
4 7
11
2
32
8
6 8
2
11 2
1 2
1 2
=+
=- +
= - +
=
=+
- =- +
- = - +
=
= =
Note that the x -coordinate of the midpoint is the average of x1 and .x2 The same applies to the y -coordinate.
PROBLEM
A timekeeper worked out the average time for 8 fi nalists in a race. The average was 30.55, but the timekeeper lost one of the fi nalist’s times. The other 7 times were 30.3, 31.1, 30.9, 30.7, 29.9, 31.0 and 30.3. Can you fi nd out the missing time?
ch7.indd 397 7/10/09 3:07:14 AM
398 Maths In Focus Mathematics Extension 1 Preliminary Course
7.2 Exercises
1. Find the midpoint of (a) ,0 2^ h and ,4 6^ h (b) ,2 3-^ h and ,4 5-^ h (c) ,2 5-^ h and ,6 7-^ h (d) ,2 3^ h and ,8 1-^ h (e) ,5 2-^ h and ,3 0^ h (f) ,2 2- -^ h and ,4 6-^ h (g) ,1 4-^ h and ,5 5^ h (h) 0, 4^ h and ,3 2-^ h (i) ,8 1-^ h and ,7 6-^ h (j) ,3 7^ h and ,3 4-^ h
2. Find the values of a and b if (a) ,4 1^ h is the midpoint of ,a b^ h
and ,1 5-^ h (b) ,1 0-^ h is the midpoint of
,a b^ h and ,3 6-^ h (c) ,a 2^ h is the midpoint of ( , b3 h
and ,5 6-^ h (d) ,2 1-^ h is the midpoint of
,a 4^ h and , b3-^ h (e) , b3^ h is the midpoint of ,a 2^ h
and ,0 0^ h 3. Prove that the origin is the
midpoint of ,3 4-^ h and ,3 4-^ h . 4. Show that P Q= where P is the
midpoint of ,2 3-^ h and ,6 5-^ h and Q is the midpoint of ,7 5- -^ h and ,11 3^ h .
5. Find the point that divides the interval between ,3 2-^ h and ,5 8^ h in the ratio of 1:1.
6. Show that the line 3x = is the perpendicular bisector of the interval between the points ,1 2-^ h and ,7 2^ h .
7. The points , , , ,A B1 2 1 5-^ ^h h ,C 6 5^ h and ,D 4 2^ h form a parallelogram. Find the midpoints of the diagonals AC and BD . What property of a parallelogram does this show?
8. The points , , , ,A B3 5 9 3-^ ^h h ,C 5 6-^ h and ,D 1 2-^ h form a quadrilateral. Prove that the diagonals are equal and bisect one another. What type of quadrilateral is ABCD ?
9. A circle with centre ,2 5-^ h has one end of a diameter at , .4 3-^ h Find the coordinates of the other end of the diameter.
10. A triangle has vertices at , ,,A B1 3 0 4-^ ^h h and ,C 2 2-^ h .
Find the midpoints (a) X , Y and Z of sides AB , AC and BC respectively.
Show that (b) ,XY BC21
=
XZ AC21
= and 21 .YZ AB=
11. Point ,x yP ^ h moves so that the midpoint between P and the origin is always a point on the circle 1.x y2 2+ = Find the equation of the locus of P .
12. Find the equation of the locus of the point ,x yP ^ h that is the midpoint between all points on the circle 4x y2 2+ = and the origin.
Gradient
The gradient of a straight line measures its slope. The gradient compares the vertical rise with the horizontal run.
The locus is the path that ( , )P x y follows.
ch7.indd 398 6/27/09 3:26:55 PM
399Chapter 7 Linear Functions
Gradient runrise
=
On the number plane, this is a measure of the rate of change of y with respect to x .
The rate of change of y with respect to x is a very important measure of their relationship. In later chapters you will use the gradient for many purposes, including sketching curves, fi nding the velocity and acceleration of objects, and fi nding maximum and minimum values of formulae.
EXAMPLES
Find the gradient of each interval. 1.
Solution
Gradient run
rise
32
=
=
You will study the gradient at different points on a curve in
the next chapter.
CONTINUED
ch7.indd 399 6/26/09 4:13:52 AM
400 Maths In Focus Mathematics Extension 1 Preliminary Course
2.
Solution
In this case, x is 3- (the run is measured towards the left) .
Gradient runrise
32
32
=
=-
= -
Positive gradient leans to the right. Negative gradient leans to the left.
Gradient given 2 points
The gradient of the line between ,x y1 1_ i and ,x y2 2_ i is given by
m x xy y
2 1
2 1=
-
-
Proof
ch7.indd 400 6/26/09 4:13:53 AM
401Chapter 7 Linear Functions
BC y y2 1= - and AC x x2 1= -
Gradient run
rise
x xy y
2 1
2 1
=
=-
-
This formula could also be
written mx x
y y
1 2
1 2=
-
-
EXAMPLES
1. Find the gradient of the line between points 2, 3^ h and , .3 4-^ h
Solution
Gradient: m x xy y
3 24 3
51
51
2 1
2 1=
-
-
=- -
-
=-
= -
2. Prove that points , ,,2 3 2 5- -^ ^h h and ,0 1-^ h are collinear.
Solution
To prove points are collinear, we show that they have the same gradient (slope).
Collinear points lie on the same line, so they have
the same gradients.
CONTINUED
ch7.indd 401 7/10/09 3:08:25 AM
402 Maths In Focus Mathematics Extension 1 Preliminary Course
Gradient of the interval between ,2 5- -^ h and ,0 1-^ h :
m x xy y
25
21 5
24
2
012 1
2 1=
-
-
=-
-
=- +
=
=
-
- -
]]gg
Gradient of the interval between ,0 1-^ h and ,2 3^ h :
m x xy y
2 01
23 1
24
2
32 1
2 1=
-
-
=-
-
=+
=
=
- ] g
Since the gradient of both intervals is the same, the points are collinear.
Gradient given the angle at the x -axis
The gradient of a straight line is given by
tanm i=
where i is the angle the line makes with the x -axis in the positive direction
Proof
runrise
adjacent
opposite
tan
m
i
=
=
=
ch7.indd 402 7/10/09 3:08:27 AM
403Chapter 7 Linear Functions
For an acute angle tan 02i . For an obtuse angle tan 01i .
Class Discussion
Which angles give a positive gradient? 1. Which angles give a negative gradient? Why? 2. What is the gradient of a horizontal line? What angle does it make 3. with the x -axis? What angle does a vertical line make with the 4. x -axis? Can you fi nd its gradient?
EXAMPLES
1. Find the gradient of the line that makes an angle of 135c with the x -axis in the positive direction.
Solution
tan
tan
m
135
1
c
i=
=
= -
2. Find the angle, in degrees and minutes, that a straight line makes with the x -axis in the positive direction if its gradient is 0.5.
Solution
.
tan
tan
m
0 5
26 34
`
c
i
i
i
=
=
= l
Can you see why the gradient is negative?
ch7.indd 403 6/26/09 4:13:57 AM
404 Maths In Focus Mathematics Extension 1 Preliminary Course
7.3 Exercises
1. Find the gradient of the line between
(a) ,3 2^ h and ,1 2-^ h (b) ,0 2^ h and ,3 6^ h (c) ,2 3-^ h and ,4 5-^ h (d) ,2 5-^ h and ,3 7-^ h (e) ,2 3^ h and ,1 1-^ h (f) ,5 1-^ h and ,3 0^ h (g) ,2 3- -^ h and ,4 6-^ h (h) ,1 3-^ h and ,7 7-^ h (i) ,1 4-^ h and ,5 5^ h (j) ,0 4^ h and ,3 2-^ h
2. If the gradient of , y8 1_ i and ,1 3-^ h is 2, fi nd the value of .y1
3. The gradient of ,2 1-^ h and ,x 0^ h is –5 . Find the value of x .
4. The gradient of a line is –1 and the line passes through the points ,4 2^ h and ,x 3-^ h . Find the value of x .
5. (a) Show that the gradient of the line through ,2 1-^ h and 3, 4^ h is equal to the gradient of the line between the points , ,and2 1 7 2-^ ^h h .(b) Draw the two lines on the number plane. What can you say about the lines?
6. Show that the points , , , , ,A B C1 2 1 5 6 5-^ ^ ^h h h and ,D 4 2^ h form a parallelogram. Find the gradients of all sides.
7. The points , , , , ,A B C3 5 9 3 5 6- -^ ^ ^h h h and
,D 1 2-^ hform a rectangle. Find the gradients of all the sides and the diagonals.
8. Find the gradients of the diagonals of the square with vertices , , , ,A B2 1 3 1-^ ^h h , and , .C D3 6 2 6-^ ^h h
9. A triangle has vertices , ,,A B3 1 1 4- -^ ^h h and , .C 11 4-^ h
By fi nding the lengths of all (a) sides, prove that it is a right-angled triangle.
Find the gradients of sides (b) AB and BC .
10. (a) Find the midpoints F and G of sides AB and AC where ABC is a triangle with vertices , ,,A B0 3 2 7-^ ^h h and ,C 8 2-^ h .(b) Find the gradients of FG and BC .
11. The gradient of the line between a moving point ,P x y^ h and the point ,A 5 3^ h is equal to the gradient of line PB where B has coordinates ,2 1-^ h . Find the equation of the locus of P .
12. Prove that the points , , ,3 1 5 5-^ ^h h and ,2 4-^ h are collinear.
13. Find the gradient of the straight line that makes an angle of 45c with the x -axis in the positive direction.
14. Find the gradient, to 2 signifi cant fi gures, of the straight line that makes an angle of 42 51c l with the x -axis.
15. Find the gradient of the line that makes an angle of 87 14c l with the x -axis, to 2 signifi cant fi gures.
16. Find the angle, in degrees and minutes, that a line with gradient 1.2 makes with the x -axis.
17. What angle, in degrees and minutes does the line with gradient –3 make with the x -axis in the positive direction?
ch7.indd 404 6/27/09 3:26:59 PM
405Chapter 7 Linear Functions
Gradient given an equation
In Chapter 5 you explored and graphed linear functions. You may have noticed a relationship between the graph and the gradient and y -intercept of a straight line.
18. Find the exact gradient of the line that makes an angle with the x -axis in the positive direction of
(a) 60c (b) 30c (c) .120c
19. Show that the line passing through ,4 2-^ h and ,7 5-^ h
makes an angle of 135c with the x -axis in the positive direction.
20. Find the exact value of x with rational denominator if the line passing through ,x 3^ h and ,2 1^ h makes an angle of 60c with the x -axis.
Investigation
1. (i) Draw the graph of each linear function. (ii) By selecting two points on the line, fi nd its gradient.
(a) y x= (b) 2y x= (c) 3y x= (d) y x= - (e) 2y x= -
Can you fi nd a pattern for the gradient of each line? Can you predict what the gradient of 5y x= and 9y x= - would be?
2. (i) Draw the graph of each linear function. (ii) Find the y -intercept.
408 Maths In Focus Mathematics Extension 1 Preliminary Course
Equation of a Straight Line
There are several different ways to write the equation of a straight line.
General form
0ax by c+ + =
Gradient form
y mx b= +
where gradientm = and b y= -intercept
Intercept form
1ax
b
y+ =
where a and b are the x -intercept and y -intercept respectively
Proof
,m ab b b
y ab x b
b
yax
ax
b
y
1
1`
`
= - =
= - +
= - +
+ =
Point-gradient formula
There are two formulae for fi nding the equation of a straight line. One of these uses a point and the gradient of the line.
The equation of a straight line is given by
x xy y m 11 -- = _ i where ,x y1 1_ i lies on the line with gradient m
This is a very useful formula as it is used in many topics in this course.
ch7.indd 408 6/26/09 4:14:03 AM
409Chapter 7 Linear Functions
Proof
Given point ,x y1 1_ i on the line with gradient m
Let ,P x y= ^ h Then line AP has gradient
m x xy y
m x xy y
m x x y y
2 1
2 1
1
1
1 1
`
=-
-
=-
-
- = -_ i
Two-point formula
The equation of a straight line is given by
x xy y
x xy y
1
1
2 1
2 1
-
-=
-
-
where ,x y1 1_ i and ,x y2 2_ i are points on the line
Proof
ABRD;
,
So
P x y
APQ
AQPQ
ARBR
x xy y
x xy y
Let
i.e.1
1
2 1
2 1
<D
=
=
-
-=
-
-
^ h
The two-point formula is not essential. The right-hand side of it is the gradient of the line. Replacing this by m gives the point–gradient formula.
This formula is optional as you can
use the point–gradient formula for any
question.
The gradient is the same anywhere along
a straight line.
ch7.indd 409 7/10/09 4:04:22 PM
410 Maths In Focus Mathematics Extension 1 Preliminary Course
EXAMPLES
1. Find the equation of the straight line with gradient 4- and passing through the point , .2 3-^ h
Solution
, andm x y4 2 31 1= - = - =
Equation: ( )
[ ( )]
( )
(gradient form)
or (general form)
y y m x x
y x
x
x
y x
x y
3 4 2
4 2
4 8
4 5
4 5 0
1 1
`
- = -
- = - - -
= - +
= - -
= - -
+ + =
2. Find the equation of the straight line that passes through the points ,2 3-^ h and , .4 7- -^ h
Solution
By two-point formula:
x xy y
x xy y
x
y
x
y
x
y
y xy x
x y
x y
4
7
2 43 7
47
2 43 7
47
32
3 7 2 43 21 2 8
2 3 13 0
2 3 13 0or
1
1
2 1
2 1
-
-=
-
-
- -
- -=
- -
- - -
+
+=
+
- +
+
+=
+ = +
+ = +
- + + =
- - =
]]
]]
^ ]
gg
gg
h g
By point-gradient method:
m x xy y
2 43 7
2 43 7
32
2 1
2 1=
-
-
=- -
- - -
=+
- +
=
]]gg
Use one of the points, say ,4 7- -^ h . , 4 7m x y
32 and1 1= = - = -
Equation: ( )
( ) ( )
y y m x x
y x732 4
1 1- = -
- - = - -6 @
ch7.indd 410 6/26/09 4:14:05 AM
411Chapter 7 Linear Functions
( )
or
y x
y xy x
x y
x y
732 4
3 7 2 43 21 2 8
2 3 13 0
2 3 13 0
`
+ = +
+ = +
+ = +
- + + =
- - =
^ ]h g
3. Find the equation of the line with x- intercept 3 and y- intercept 2.
Solution
Intercept form is 1,ax
b
y+ = where a and b are the x- intercept and
y- intercept respectively.
1
2 3 6
2 3 6 0
x y
x y
x y
3 2`
`
+ =
+ =
+ - =
Again, the point-gradient formula can be used. The x -intercept and y -intercept are the points 3, 0^ h and , .0 2^ h
7.5 Exercises
1. Find the equation of the straight line
with gradient 4 and (a) y -intercept 1-
with gradient (b) 3- and passing through ,0 4^ h
passing through the origin (c) with gradient 5
with gradient 4 and (d) x -intercept 5-
with (e) x -intercept 1 and y -intercept 3
with (f) x -intercept 3, y -intercept 4-
with (g) y -intercept 1- and making an angle of 45c with the x -axis in the positive direction
with (h) y -intercept 5 and making an angle of 45c with the x -axis in the positive direction.
2. Find the equation of the straight line that makes an angle of 135c with the x -axis and passes through the point , .2 6^ h
3. Find the equation of the straight line passing through
(a) ,2 5^ h and ,1 1-^ h (b) ,0 1^ h and ,4 2- -^ h (c) ,2 1-^ h and ,3 5^ h (d) ,3 4^ h and ,1 7-^ h (e) ,4 1- -^ h and ,2 0-^ h .
4. What is the equation of the line with x -intercept 2 and passing through , ?3 4-^ h
5. Find the equation of the line parallel to the (a) x -axis and
passing through ,2 3^ h parallel to the (b) y -axis and
passing through ,1 2-^ h .
ch7.indd 411 7/10/09 3:09:40 AM
412 Maths In Focus Mathematics Extension 1 Preliminary Course
Parallel and Perpendicular Lines
Parallel lines
6. A straight line passing through the origin has a gradient of 2.- Find its equation.
7. A straight line has x -intercept 4 and passes through , .0 3-^ h Find its equation.
8. Find the equation of the straight line with gradient 2- that passes through the midpoint of ,5 2-^ h and , .3 4-^ h
9. What is the equation of the straight line through the point ,4 5-^ h and the midpoint of ,1 2^ h and , ?9 4-^ h
10. What is the equation of the straight line through the midpoint of ,0 1^ h and ,6 5-^ h and the midpoint of ,2 3^ h and , ?8 3-^ h
Class Investigation
Sketch the following straight lines on the same number plane . 1. y x2=
y x2 1= +2. y x2 3= -3. y x2 5= +4.
What do you notice about these lines?
If two lines are parallel, then they have the same gradient. That is, m m1 2=
Two lines that are parallel have equations 0ax by c1+ + = and 0ax by c2+ + =
ch7.indd 412 7/10/09 3:09:41 AM
413Chapter 7 Linear Functions
Proof
0ax by c1+ + = has gradient mba
1 = -
0ax by c2+ + = has gradient mba
2 = -
Since ,m m1 2= the two lines are parallel.
EXAMPLES
1. Prove that the straight lines 5 2 1 0x y- - = and 5 2 7 0x y- + = are parallel.
Solution
x y
x y
x y
m
x y
x y
x y
m
5 2 1 0
5 1 2
25
21
25
5 2 7 0
5 7 2
25
27
25
1
2
`
`
- - =
- =
- =
=
- + =
+ =
+ =
=
25m m1 2= =
` the lines are parallel .
2. Find the equation of a straight line parallel to the line 2 3 0x y- - = and passing through , .1 5-^ h
Solution
2 3 0
2 3
2
x y
x y
m1`
- - =
- =
=
For parallel lines m m1 2= 2m2` =
Equation: ( )( ) ( )
y y m x xy x
y xx y
5 2 15 2 20 2 7
1 1- = -
- - = -
+ = -
= - -
Notice that the equations are both in the form
5x 2y k 0.- + =
ch7.indd 413 6/26/09 4:14:07 AM
414 Maths In Focus Mathematics Extension 1 Preliminary Course
DID YOU KNOW?
Parallel lines are usually thought of as lines that never meet. However, there is a whole branch of geometry based on the theory that parallel lines meet at infi nity. This is called affi ne geometry . In this geometry there are no perpendicular lines.
Perpendicular lines
Class Investigation
Sketch the following pairs of straight lines on the same number plane.
If two lines with gradients m1 and m2 respectively are perpendicular, then
1m m
m m1i.e.
1 2
21
= -
= -
Proof
Let line AB have gradient tanm1 a= . Let line CD have gradient tanm2 b= .
straight angle
tan
tan
cot
ECEB
CBE
EBEC
ECEB
180
180
180`
c
c
c
+
b
a
a
a
=
= -
- =
- =
^]]
hgg
Gradients of perpendicular lines are negative reciprocals of each other.
ch7.indd 414 6/26/09 4:14:08 AM
415Chapter 7 Linear Functions
So
or
tan cotcot
tan
m mm m
180
1
1
1
21
1 2
` cb a
a
a
= -
= -
= -
= -
= -
] g
Perpendicular lines have equations in the form 0ax by c1+ + = and 0bx ay c2- + =
Proof
has gradient
has gradient
ax by c mba
bx ay c m ab
ab
0
0
1 1
2 2
+ + = = -
- + = = --
=
m mba
ab
1
1 2 #= -
= -
Since ,m m 11 2 = - the two lines are perpendicular .
EXAMPLES
1. Show that the lines 3 11 0x y+ - = and 3 1 0x y- + = are perpendicular.
Solution
3
1
x y
y x
m
x y
x y
x y
m
m m
3 11 0
3 11
3
3 1 0
1 3
31
31
31
31
1
2
1 2
`
`
#
+ - =
= - +
= -
- + =
+ =
+ =
=
= -
= -
the lines are perpendicular .
Notice that the equations are in the form
x y c3 01+ + = and .x y c3 02- + =
CONTINUED
ch7.indd 415 6/26/09 4:14:09 AM
416 Maths In Focus Mathematics Extension 1 Preliminary Course
2. Find the equation of the straight line through 2, 3^ h perpendicular to the line that passes through ,1 7-^ h and , .3 3^ h
Solution
Line through ,1 7-^ h and , :3 3^ h
1
m x xy y
m1 3
7 3
44
2 1
2 1
1
=-
-
=- -
-
=-
= -
For perpendicular lines, 1= -m m1 2
1 1m
m 1
i.e. 2
2
- = -
=
Equation through , :2 3^ h
( )
( )
y y m x x
y x
x
x y
3 1 2
2
0 1
1 1- = -
- = -
= -
= - +
7.6 Exercises
1. Find the gradient of the straight line
parallel to the line (a) 3 4 0x y+ - =
perpendicular to the line (b) 3 4 0x y+ - =
parallel to the line joining (c) ,3 5^ h and ,1 2-^ h
perpendicular to the line with (d) x -intercept 3 and y -intercept 2
perpendicular to the line (e) making an angle of 135c with the x -axis in the positive direction
perpendicular to the line (f) 6 5 4 0x y- - =
parallel to the line making an (g) angle of c30 with the x -axis
parallel to the line (h) 3 7 0x y- - =
perpendicular to the line (i) making an angle of c120 with the x -axis in the positive direction
perpendicular to the line (j) passing through ,4 2-^ h and , .3 3^ h
2. Find the equation of each straight line
passing through (a) ,2 3^ h and parallel to the line 6y x= +
through (b) ,1 5-^ h and parallel to the line 3 7 0x y- - =
with (c) x -intercept 5 and parallel to the line 4y x= -
through (d) ,3 4-^ h and perpendicular to the line 2y x=
through (e) ,2 1-^ h and perpendicular to the line 2 3 0x y+ + =
ch7.indd 416 6/26/09 4:14:10 AM
417Chapter 7 Linear Functions
through (f) ,7 2-^ h and perpendicular to the line 3 5 0x y- - =
through (g) ,3 1- -^ h and perpendicular to the line .x y4 3 2 0- + =
3. Show that the straight lines 3 2y x= - and 6 2 9 0x y- - = are parallel .
4. Show that lines 5 0x y+ = and 5 3y x= + are perpendicular .
5. Show that lines 6 5 1 0x y- + = and 6 5 3 0x y- - = are parallel.
6. Show that lines 7 3 2 0x y+ + = and 3 7 0x y- = are perpendicular.
7. If the lines 3 2 5 0x y- + = and 1y kx= - are perpendicular, fi nd the value of k .
8. Show that the line joining ,3 1-^ h and ,2 5-^ h is parallel to the line 8 2 3 0.x y- - =
9. Show that the points , ,A 3 2- -^ h , ,B 1 4-^ h , ,C 7 1-^ h and ,D 5 7-^ h are the vertices of a parallelogram.
10. The points , ,A 2 0-^ h , ,B 1 4^ h ,C 6 4^ h and ,D 3 0^ h form a rhombus. Show that the diagonals are perpendicular.
11. Find the equation of the straight line
passing through the (a) origin and parallel to the line 3 0x y+ + =
through (b) ,3 7^ h and parallel to the line 5 2 0x y- - =
through (c) ,0 2-^ h and perpendicular to the line 2 9x y- =
perpendicular to the line (d) 3 2 1 0x y+ - = and passing through the point ,2 4-^ h .
12. Find the equation of the straight line passing through ,6 3-^ h that is perpendicular to the line joining ,2 1-^ h and , .5 7- -^ h
13. Find the equation of the line through ,2 1^ h that is parallel to the line that makes an angle of c135 with the x -axis in the positive direction.
14. Find the equation of the perpendicular bisector of the line passing through ,6 3-^ h and , .2 1-^ h
15. Find the equation of the straight line parallel to the line 2 3 1 0x y- - = and through the midpoint of ,1 3^ h and , .1 9-^ h
Intersection of Lines
Two straight lines intersect at a single point , .x y^ h The point satisfi es the equations of both lines. We fi nd this point by solving simultaneous equations.
You may need to revise simultaneous equations
from Chapter 3 .
ch7.indd 417 6/26/09 4:14:11 AM
418 Maths In Focus Mathematics Extension 1 Preliminary Course
Concurrent lines meet at a single point. To show that lines are concurrent, solve two simultaneous equations to fi nd the point of intersection. Then substitute this point of intersection into the third and subsequent lines to show that these lines also pass through the point.
EXAMPLES
1. Find the point of intersection between lines x y3 3 02 - - = and .x y5 2 13 0- - =
Solution
Solve simultaneous equations:
:
:
:
x
x y
x y
x y
x y
xx
3
2 3 3 0 1
5 2 13 0 2
1 2 4 6 6 0 3
2 3 15 6 39 0 4
3 4 11 33 033 11
#
#
=
- - =
- - =
- - =
- - =
- - + =
=
^^
^ ^^ ^^ ^
hh
h hh hh h
ubstitute into :S x 3 1= ^ hyy
y
y
2 3 3 3 03 3 0
3 3
1
- - =
- + =
=
=
^ h
So the point of intersection is , .3 1^ h
2. Show that the lines ,x y x y3 1 0 2 12 0- + = + + = and x y4 3 7 0- - = are concurrent.
Solution
Solve any two simultaneous equations:
:
:
x y
x y
x y
x y
x
3 1 0 1
2 12 0 2
4 3 7 0 3
1 2 6 2 2 0 4
2 4 7 14 0
#
- + =
+ + =
- - =
- + =
+ + =
^^^
^ ^^ ^
hhh
h hh h
You could use a computer spreadsheet to solve these simultaneous equations.
ch7.indd 418 7/10/09 3:09:42 AM
419Chapter 7 Linear Functions
7 14
x
x
2= -
= -
ubstitute into :xS 2 1= - ^ hyy
y
3 2 1 05 0
5
- - + =
- - =
- =
^ h
So the point of intersection of (1) and (2) is ,2 5- -^ h . Substitute ,2 5- -^ h into (3): x y4 3 7 0- - =
LHS 4 2
RHS
3 5 7
8 15 7
0
= - - -
= - + -
=
=
-^ ^h h
So the point lies on line (3) all three lines are concurrent .
Equation of a line through the intersection of 2 other lines
To fi nd the equation of a line through the intersection of 2 other lines, fi nd the point of intersection, then use it with the other information to fi nd the equation.
Another method uses a formula to fi nd the equation.
If a x b y c 01 1 1+ + = and a x b y c 02 2 2+ + = are 2 given lines then the equation of a line through their intersection is given by the formula
( ) ( )a x b y c k a x b y c 01 1 1 2 2 2+ + + + + = where k is a constant
Proof
Let l1 have equation .a x b y c 01 1 1+ + = Let l2 have equation .a x b y c 02 2 2+ + = Let the point of intersection of l1 and l2 be ,x yP 1 1^ h . Then P satisfi es l1
i.e. a x b y c 01 11 1 1+ + =
P also satisfi es l2
i.e. a x b y c 01 12 2 2+ + =
Substitute P into ( ) ( )a x b y c k a x b y c 01 1 1 2 2 2+ + + + + =
( ) ( )a x b y c k a x b y c
k
0
0 0 00 0
1 1 1 11 1 1 2 2 2+ + + + + =
+ =
=
^ h if point P satisfi es both equations l1 and l2 then it satisfi es l kl 01 2+ = .
ch7.indd 419 6/26/09 4:14:13 AM
420 Maths In Focus Mathematics Extension 1 Preliminary Course
7.7 Exercises
EXAMPLE
Find the equation of the line through ,1 2-^ h that passes through the intersection of lines x y2 5 0+ - = and .x y3 1 0- + =
Solution
Using the formula: , , , ,a b c a b c2 1 5 1 3 11 1 1 2 2 2= = = - = = - =
a x b y c k a x b y c
x y k x y
0
2 5 3 1 01 1 1 2 2 2+ + + + + =
+ - + - + =
^ ^^ ^
h hh h
Since this line passes through , ,1 2-^ h substitute the point into the equation:
kk
k
k
2 2 5 1 6 1 05 6 0
5 6
65
- + - + - - + =
- - =
- =
- =
^ ^h h
So the equation becomes:
x y x y
x y x yx y x y
x y
x y
2 565 3 1 0
6 2 5 5 3 1 012 6 30 5 15 5 0
7 21 35 0
3 5 0
+ - - - + =
+ - - - + =
+ - - + - =
+ - =
+ - =
^ ^^ ^
h hh h
Another way to do this example is to fi nd the point of intersection, then use both points to fi nd the equation.
Substitute the value of k back into the equation.
1. Find the point of intersection of straight lines
(a) x y3 4 10 0+ + = and x y2 3 16 0- - =
x y5 2 11 0+ + =(b) and x y3 6 0+ + =
x y7 3 16- =(c) and x y5 2 12- =
x y2 3 6- =(d) and x y4 5 10- =
x y3 8 0- - =(e) and x y4 7 13 0+ - =
y x5 6= +(f) and y x4 3= - -
y x2 1= +(g) and x y5 3 6 0+ =-
x y3 7 12+ =(h) and x y4 1 06- =-
x y3 5 7- = -(i) and x y2 3 4- =
x y8 7 3 0- - =(j) and x y5 2 1 0- - =
2. Show that the lines x y2 11 0- - = and
x y2 10 0- =- intersect at the point , .3 4-^ h
3. A triangle is formed by 3 straight lines with equations ,x y2 1 0- + = x y2 09+ - =
ch7.indd 420 6/27/09 3:27:00 PM
421Chapter 7 Linear Functions
and .x y2 5 3 0- - = Find the coordinates of its vertices.
4. Show that the lines ,x y5 17 0- - =
x y3 2 12 0- - = and x y5 7 0+ - = are concurrent.
5. Show that the lines ,x y4 5 0+ + = ,x y3 7 15 0- + =
x y2 10 0- =+ and x y6 5 30 0+ + = are concurrent.
6. Find the equation of the straight line through the origin that passes through the intersection of the lines x y5 2 14 0- + = and
x y3 4 7 0+ - = .
7. Find the equation of the straight line through ,3 2^ h that passes through the intersection of the lines x y5 2 01+ + = and
x y3 16 0- + = .
8. Find the equation of the straight line through ,4 1- -^ h that passes through the intersection of the lines x y2 1 0+ - = and .x y3 5 16 0+ + =
9. Find the equation of the straight line through ,3 4-^ h that passes through the intersection of the lines x y2 3 0+ - = and x y3 2 8 0- - = .
10. Find the equation of the straight line through ,2 2-^ h that passes through the intersection of the lines x y2 3 6 0+ - = and x y3 5 10 0+ =- .
11. Find the equation of the straight line through ,3 0^ h that passes through the intersection of the lines x y 1 0- + = and x y4 2 0- - = .
12. Find the equation of the straight line through ,1 2- -^ h that passes through the intersection of the lines x y2 6 0+ - = and .x y3 7 9 0+ - =
13. Find the equation of the straight line through ,1 2^ h that passes through the intersection of the lines x y2 10 0+ + = and .x y 02 5- =+
14. Find the equation of the straight line through ,2 0-^ h that passes through the intersection of the lines x y3 4 7 0+ - = and .x y3 2 1 0- - =
15. Find the equation of the straight line through ,3 2-^ h that passes through the intersection of the lines x y5 2 13 0+ - = and x y3 11 0- + = .
16. Find the equation of the straight line through ,3 2- -^ h that passes through the intersection of the lines x y 1 0+ + = and x y3 2 0+ = .
17. Find the equation of the straight line through ,3 1^ h that passes through the intersection of the lines x y3 4 0- + = and x y2 12 0- + = .
18. Find the equation of the straight line with gradient 3 that passes through the intersection of the lines x y2 1 0+ - = and x y3 5 16 0+ + = .
19. Find the equation of the straight line with gradient 2 that passes through the intersection of the lines x y5 2 3 0- - = and x y7 3 4 0- - = .
ch7.indd 421 6/26/09 4:14:15 AM
422 Maths In Focus Mathematics Extension 1 Preliminary Course
20. Find the equation of the straight line parallel to the line x y3 7 0- - = that passes through the intersection of the lines x y3 2 10 0- - = and
.x y4 17 0+ - =
21. Find the equation of the straight line perpendicular to the line x y5 1 0+ - = that passes through the intersection of lines x y3 5 3 0- - = and x y2 3 17 0+ + = .
Perpendicular Distance
The distance formula d x x y y2 12
2 12= - + -_ _i i is used to fi nd the distance
between two points. Perpendicular distance is used to fi nd the distance between a point and
a line. If we look at the distance between a point and a line, there could be many distances.
So we choose the shortest distance, which is the perpendicular distance.
The perpendicular distance from ,x y1 1_ i to the line 0ax by c+ + = is
given by | |
da b
ax by c2 2
1 1=
+
+ +
A distance is always positive, so take the absolute value.
Proof
ch7.indd 422 6/26/09 4:14:16 AM
423Chapter 7 Linear Functions
Let d be the perpendicular distance of ,x y1 1_ i from the line .ax by c 0+ + =
,A ac 0= -b l ,C
bc0= -c m ,R x
b
ax c1
1=
- -e o
,ACOac
bc
a bc b c a
abc a b
In AC2
2
2
2
2 2
2 2 2 2
2 2
D = +
=+
=+
PR yb
ax c
b
ax by c
1
1
1 1
= -- -
=+ +
e o
ACOD is similar to PRQD
.AOPQ
ACPR
PQAC
AO PR
d ac
b
ax by c
abc a b
ab
c ax by c
c a b
ab
a b
ax by c
1 12 2
1 1
2 2
2 2
1 1
`
` # '
#
=
=
=+ + +
=+ +
+
=+
+ +
_ i
All points on one side of the line 0ax by c+ + = make the numerator of this formula positive. Points on the other side make the numerator negative.
Usually we take the absolute value of d . However, if we want to know if points are on the same side of a line or not, we look at the sign of d .
To fi nd A and C , substitute y 0= and x 0= into
.ax by c 0+ + =
Why?
EXAMPLES
1. Find the perpendicular distance of ,4 3-^ h from the line .x y3 4 1 0- - =
Solution
, , , ,
| |
| |
x y a b c
da b
ax by c
4 3 3 4 1
3 4
3 4 4 3 1
1 1
2 2
1 1
2 2
= = - = = - = -
=+
+ +
=+ -
+ - - + -
]] ] ] ]
gg g g g
CONTINUED
ch7.indd 423 7/10/09 3:10:25 AM
424 Maths In Focus Mathematics Extension 1 Preliminary Course
| |
.
25
12 12 1
523
4 6
=+ -
=
=
So the perpendicular distance is 4.6 units.
2. Prove that the line x y6 8 20 0+ + = is a tangent to the circle 4.x y2 2+ =
Solution
There are three possibilities for the intersection of a circle and a straight line.
The centre of the circle x y 42 2+ = is ,0 0^ h and its radius is 2 units. A tangent is perpendicular to the centre of the circle. So we prove that the
perpendicular distance from the line to the point ,0 0^ h is 2 units (the radius).
| |
| ( ) ( ) |
| |
da b
ax by c
6 8
6 0 8 0 20
100
20
1020
2
2 2
1 1
2 2
=+
+ +
=+
+ +
=
=
=
the line is a tangent to the circle.
3. Show that the points ,1 3-^ h and ,2 7^ h lie on the same side of the line .x y2 3 4 0- + =
ch7.indd 424 6/26/09 4:14:18 AM
425Chapter 7 Linear Functions
Solution
To show that points lie on the same side of a line, their perpendicular distance must have the same sign. We use the formula without the absolute value sign.
, :
, :
da b
ax by c
d
d
1 3
2 3
2 1 3 3 4
4 92 9 4
137
2 7
2 3
2 2 3 7 4
4 94 21 4
1313
2 2
1 1
2 2
2
=+
+ +
-
=+ -
- - +
=+
- - +
=-
=+ -
- +
=+
- +
=-
2
^]
] ]
^]
] ]
hg
g g
hg
g g
Since the perpendicular distance for both points has the same sign, the points lie on the same side of the line.
1. Find the perpendicular distance between
(a) ,1 2^ h and x y3 4 2 0+ + = (b) ,3 2-^ h and 5 12 7 0x y+ + = (c) ,0 4^ h and 8 6 1 0x y- - = (d) 3, 2- -^ h and x y4 3 6 0- - = the origin and (e)
.x y12 5 8 0- + =
2. Find, correct to 3 signifi cant fi gures, the perpendicular distance between
(a) ,1 3^ h and 3 1 0x y+ + = (b) ,1 1-^ h and 2 5 4 0x y+ + = (c) ,3 0^ h and 5 6 12 0x y- - = (d) ,5 3-^ h and 4 2 0x y- - = (e) 6, 3- -^ h and .x y2 3 9 0- + =
3. Find as a surd with rational denominator the perpendicular distance between
the origin and the line (a) 3 2 7 0x y- + =
(b) ,1 4-^ h and 2 3 0x y+ + = (c) ,3 1-^ h and 3 14 1 0x y+ + = (d) 2, 6-^ h and 5 6 0x y- - = (e) 4, 1- -^ h and
.x y3 2 4 0- - =
4. Show that the origin is equidistant from the lines 7 24 25 0,x y+ + = 4 3 5 0x y+ - = and 12 5 13 0.x y+ - =
7.8 Exercises
ch7.indd 425 6/26/09 4:14:19 AM
426 Maths In Focus Mathematics Extension 1 Preliminary Course
Equidistant means that two or more objects are the same distance away from another object.
5. Show that points ,A 3 5-^ h and ,B 1 4-^ h lie on opposite sides of 2 3 0.x y- + =
6. Show that the points 2, 3-^ h and ,9 2^ h lie on the same side of the line .x y3 2 0- + =
7. Show that 3, 2-^ h and ,4 1^ h lie on opposite sides of the line .x y4 3 2 0- - =
8. Show that 0, 2-^ h is equidistant from the lines 3 4 2 0x y+ - = and .x y12 5 16 0- + =
9. Show that the points 8, 3-^ h and ,1 1^ h lie on the same side of the line .x y6 4 0- + =
10. Show that 3, 2-^ h and ,4 1^ h lie on opposite sides of the line .x y2 2 0+ - =
11. Show that the point ,3 2-^ h is the same distance from the line 6 8 6 0x y- + = as the point 4, 1- -^ h is from the line .x y5 12 20 0+ - =
12. Find the exact perpendicular distance with rational denominator from the point ,4 5^ h to the line with x -intercept 2 and y -intercept .1-
13. Find the perpendicular distance from ,2 2-^ h to the line passing through ,3 7^ h and , .1 4-^ h
14. Find the perpendicular distance between ,0 5^ h and the line through ,3 8-^ h parallel to 4 3 1 0.x y- - =
15. The perpendicular distance between the point , 1x -^ h and the line 3 4 7 0x y- + = is 8 units. Find two possible values of x .
16. The perpendicular distance between the point , b3^ h and the line 5 12 2 0x y- - = is 2 units. Find the values of b .
17. Find m if the perpendicular distance between ,m 7^ h and the line 9 12 6 0x y+ + = is 5 units.
18. Prove that the line 3 4 25 0x y- + = is a tangent to the circle with centre the origin and radius 5 units.
19. Show that the line 3 4 12 0x y- + = does not cut the circle 1.x y2 2+ =
20. The sides of a triangle are formed by the lines with equations 2 7 0, 3 5 4 0x y x y- - = + - = and 3 4 0.x y+ - =
Find the vertices of the (a) triangle.
Find the exact length of all (b) the altitudes of the triangle.
Angle Between Two Lines
The acute angle i between two straight lines is given by
1
tanm m
m m
1 2
1 2i =
+
-
where m1 and m2 are the gradients of the lines
ch7.indd 426 7/10/09 4:04:42 PM
427Chapter 7 Linear Functions
Proof
Let line l1 have gradient m1 and line l2 have gradient .m2 Then tanm1 b= and tanm2 a=
( )tan
tan tan
tan tan
tan
ABC
m m
m m1
1
exterior angle of
1 2
1 2
`
b a i
i b a
i b a
b a
b a
D= +
= -
= -
=+
-
=+
-
^ h
When tan i is positive, i is acute. When tan i is negative, i is obtuse. for the acute angle between lines l1 and ,l2
1
tanm m
m m
1 2
1 2i =
+
-
Note: the denominator cannot be zero, so
.mm 121! - So this
formula doesn’t work for perpendicular lines.
EXAMPLES
1. Find the acute angle between the lines x y3 2 1 0- + = and .x y3 0- =
Solution
tan
x y
x y
x y
m
x y
x y
x y
m
m m
m m
3 2 1 0
3 1 2
23
21
23
3 0
3
31
31
1
So
So
1
2
1 2
1 2i
- + =
+ =
+ =
=
- =
=
=
=
=+
-
CONTINUED
ch7.indd 427 7/10/09 4:05:05 PM
428 Maths In Focus Mathematics Extension 1 Preliminary Course
tan
123
31
23
31
97
97
37 52
1
#
c
i
=
+
-
=
=
=
-
l
c m
2. Find the obtuse angle between the lines 5 2 6 0x y- + = and .x y2 4 0+ - =
Solution
3
x y
x y
x y
5 2 6 0
5 6 2
25
- + =
+ =
+ =
m25So 1 =
x y
y x
2 4 0
2 4
+ - =
= - +
m 2So 2 = -
°
tan
tan
m m
m m
1
125 2
25 2
89
89
89
48 22
1 2
1 2
1
#
l
i
i
=+
-
=
+ -
- -
= -
=
=
=
-
]]
c
gg
m
This gives the acute angle.
180 48 22131 38
Obtuse angle c c lc l
= -
=
3. If the angle between the lines 2 7 0x y- - = and 3y mx= + is ,25c fi nd two possible values of m , correct to 1 decimal place.
Solution
( )x y
x y
m
2 7 0 1
2 7
21`
- - =
- =
=
Notice that -tan8
91- d n gives 48c- 22l so we need to fi nd the obtuse angle by subtracting the acute angle from 180c .
ch7.indd 428 6/26/09 4:14:23 AM
429Chapter 7 Linear Functions
( )
°
tan
tan
y mx
m m
m m
m m
mm
3 2
1
251 22
2
1 2
1 2
`
i
= +
=
=+
-
=+
-
There are two possibilities:
(1)
( )
( )
.
tan
tan
tan tan
tan tan
tan tan
tantan
mm
m m
m m
m m
m
m
251 22
25 1 2 2
25 2 25 2
2 25 2 25
2 25 1 2 25
2 25 12 25
0 8
c
c
c c
c c
c c
c
c
Z
=+
-
+ = -
+ = -
+ = -
+ = -
=+
-
(2)
( )
( )
.
tan
tan
tan tan
tan tan
tan tan
tantan
mm
m m
m m
m m
m
m
251 22
25 1 2 2
25 2 25 2
2 25 2 25
2 25 1 2 25
2 25 12 25
36 6
c
c
c c
c c
c c
c
c
Z
- =+
-
- + = -
- - = -
- + = +
- + = +
=- +
+
1. Find the acute angle between the lines
(a) x y2 1 0+ + = and 4 0x y+ + =
(b) 3 7 0x y- - = and 5 3 0x y+ + =
(c) 2 0x y+ = and 3 2 1 0x y- + =
(d) 3 2 0x y+ + = and 4 4 1 0x y+ - =
(e) 2 5 3 0x y- - = and 5 0x y- =
(f) 3 1 0x y+ + = and 4 7 2 0x y+ + =
(g) 2 7 1 0x y- - = and 3 2 4 0x y+ - =
(h) 2 2 1 0x y+ + = and 2 4x y+ =
(i) 3 4 1 0x y+ + = and 5 2 2 0x y- - =
(j) 2 3 0x y- - = and .x y6 3 4 0- + =
2. Find the obtuse angle between the lines
(a) 4 2 0x y+ + = and 1 0x y+ - =
(b) 2 3 9 0x y- - = and 2 4 0x y+ + =
7.9 Exercises
ch7.indd 429 6/26/09 4:14:24 AM
430 Maths In Focus Mathematics Extension 1 Preliminary Course
(c) 6 2x y+ = and 2 4 3 0x y- + =
(d) 5 2 1 0x y+ + = and 4 7 0x y+ - =
(e) 4 2 7 0x y- - = and .x y3 0- =
3. Find the acute angle between the line 2 5 1 0x y- + = and the line joining ,1 2-^ h and , .5 3^ h
4. Find the acute angle between the line joining ,3 2^ h and ,1 4-^ h and the line joining ,0 5^ h and , .2 7-^ h
5. , ,A 2 1-^ h ,B 3 4-^ h and ,C 1 5-^ h form the vertices of a triangle. Find the interior angles of the triangle.
6. Find two possible values of m if the lines 2 5 0x y+ - = and 1y mx= + intersect at an angle of .45c
7. Lines 2y mx= + and 5 9y x= - intersect at an acute angle whose
tangent is 52 . Find the possible
values of m .
8. Find the values of k if the angle between the lines 6 3 4 0x y- - = and 5 0kx y- + = is .58c
9. , , ,, ,A B C0 0 1 2 5 2^ ^ ^h h h and ,D 4 0^ h form the vertices of a parallelogram.
By fi nding all the interior (a) angles, show that opposite angles are equal.
Find the obtuse angle (b) between the diagonals of the parallelogram.
10. By calculating the interior angles, show that ABCD with vertices , ,,A B7 1 1 1- -^ ^h h and ,C 5 7-^ h is an isosceles triangle.
The coordinates of a point P that divides the interval between points ,x y1 1_ i and ,x y2 2_ i in the ratio :m n respectively are given by
x m nmx nx2 1
=+
+ and y m n
my ny2 1=
+
+
Proof
Ratios
You have a formula for the midpoint which divides an interval in half. Sometimes we may want to divide an interval into a ratio that is not a half. Here is a formula that we can use to divide an interval into any internal or external ratio.
ch7.indd 430 7/10/09 3:10:29 AM
431Chapter 7 Linear Functions
Let ,P x y^ h be the point dividing the interval AB into the ratio : .m n
Then PBAP
nm
=
Draw ADC parallel to the x -axis. Then AD x x1= - and .DC x x2= -
PD BC<
PBAP
DCAD
nm
x xx x
x x x xmx mx nx nx
mx nx mx nx
m n
m nmx nx
x
m n
x
intercepts have equal ratios
2
1
2 1
2 1
2 1
2 1
`
`
=
=-
-
- -
- = -
+ = +
+
+
+=
=
=
^
_ _
]
h
i i
g
Similarly, by drawing AEF perpendicular to the x -axis, we can show that
.y m nmy ny2 1
=+
+
If P divides the interval internally in the ratio : ,m n then the ratio is positive and P lies on AB .
If P divides the interval externally in the ratio : ,m n then the ratio is negative and P lies outside AB .
A ratio of :1 1 gives the midpoint
,xx x
21 2
=+
.yy y
21 2
=+
m and n are measured in opposite directions so they
have opposite signs.
EXAMPLES
1. Divide AB into the ratio :3 4 where A is ,6 2-^ h and B is , .7 5-^ h
Solution
CONTINUED
ch7.indd 431 6/26/09 4:14:26 AM
432 Maths In Focus Mathematics Extension 1 Preliminary Course
,
x m nmx nx
y m nmy ny
P
3 43 7 4 6
73
3 43 5 4 2
77
1
73 1
2 1
2 1
`
=+
+
=+
- +
=
=+
+
=+
+ -
=
=
=
] ]
] ]
c
g g
g g
m
2. If A is ,2 1- -^ h and B is , ,1 5^ h fi nd the coordinates of the point P that divides AB externally in the ratio : .2 5
Solution
Let the ratio be : .2 5-
( )( ) [ ( )]
x m nmx nx
2 52 1 5 2
312
4
2 1=
+
+
=+ -
+ - -
=-
= -
( )( ) [ ( )]
5
,
y m nmy ny
P
2 52 5 5 1
315
4 5
2 1
`
=+
+
=+ -
+ - -
=-
= -
= - -^ h
You could use :2 5- instead and would still get the same answer for P.
ch7.indd 432 6/26/09 4:14:28 AM
433Chapter 7 Linear Functions
1. Divide these intervals internally. (a) ,1 5-^ h and ,0 4-^ h in the
ratio :2 3 (b) ,3 2-^ h and ,2 5^ h in the
ratio :4 1 (c) ,3 3-^ h and ,2 1-^ h in the
ratio :5 4 (d) ,3 1-^ h and ,7 2-^ h in the
ratio :2 5 (e) ,2 1-^ h and ,5 4-^ h in the
ratio :7 3 (f) ,2 0-^ h and ,6 3-^ h in the
ratio :3 1 (g) ,4 9^ h and ,4 1-^ h in the
ratio :1 6 (h) ,3 0-^ h and ,5 6- -^ h in the
ratio :2 9 (i) ,2 5^ h and ,3 1- -^ h in the
ratio :4 3 (j) ,1 1^ h and ,3 7-^ h in the
ratio : .1 2
2. Divide these intervals externally. (a) ,2 3-^ h and ,6 1^ h in the
ratio :1 5 (b) ,4 0^ h and ,3 5- -^ h in the
ratio :2 7 (c) ,1 1-^ h and ,4 7^ h in the
ratio :4 3 (d) ,0 2-^ h and ,8 3^ h in the
ratio :3 1 (e) ,5 2-^ h and ,4 4^ h in the
ratio :5 4 (f) ,7 1-^ h and ,0 1^ h in the
ratio :2 9 (g) ,2 2-^ h and ,6 7^ h in the
ratio :1 3 (h) ,1 3^ h and ,7 2^ h in the ratio :4 1 (i) ,4 0-^ h and ,5 5-^ h in the
ratio :6 7 (j) ,2 3-^ h and ,7 7^ h in the
ratio : .8 3
3. , ,,A B0 0 1 3^ ^h h and ,C 3 0^ h are the vertices of a triangle.
Find the coordinates of point (a) E , which divides AB internally in the ratio : .2 1
Find the coordinates of point (b) F , which divides CB internally in the ratio : .2 1
Hence prove that (c) 3 .AC EF=
4. Divide the interval AB where ,A 3 2= ^ h and ,B 1 6= -^ h into three equal parts.
5. A has coordinates ,2 5-^ h and B has coordinates , .4 3-^ h Find the length of PQ if P divides AB internally in the ratio :3 2 and Q divides AB externally in the ratio : .3 2
6. An interval AB is divided internally at P in the ratio : .5 4 If A is ,1 2-^ h and P is , ,5 6-^ h fi nd the coordinates of B .
7. The point ,5 5^ h divides the interval between , p1-^ h and ,q 6^ h in the ratio : .2 5 Find the value of p and q .
8. A triangle is formed with vertices , ,,A B5 6 0 4-^ ^h h and , .C 3 3-^ h
Find the point of intersection (a) of its medians.
If (b) D , E and F are the midpoints of AB , AC and BC , divide the intervals CD , BE and AF in the ratio : .2 1 What property of medians does this show?
9. If ,0 0^ h divides the interval AB where ,A a b= ^ h and ,B 4 9= ^ h in the external ratio of : ,2 1 fi nd the value of a and b .
10. P divides the interval between the point ,2 3^ h and the intersection of lines 2 3 19 0x y- + = and 5 2 0x y+ = in the ratio of : .4 5 Find the coordinates of P .
7.10 Exercises
ch7.indd 433 6/26/09 4:14:29 AM
434 Maths In Focus Mathematics Extension 1 Preliminary Course
Test Yourself 7 1. Find the distance between points ,1 2-^ h
and , .3 7^ h
2. What is the midpoint of the origin and the point , ?5 4-^ h
3. Find the gradient of the straight line passing through (a) ,3 1-^ h and ,2 5-^ h with equation (b) 2 1 0x y- + = making an angle of (c) 30c with the
x -axis in the positive direction perpendicular to the line (d)
.x y5 3 8 0+ - =
4. Find the equation of the linear function passing through (a) ,2 3^ h and with
gradient 7 parallel to the line (b) 5 3 0x y+ - =
and passing through ,1 1^ h through the origin, and (c)
perpendicular to the line 2 3 6 0x y- + = through (d) ,3 1^ h and ,2 4-^ h with (e) x -intercept 3 and y -intercept – .1
5. Find the perpendicular distance between ,2 5^ h and the line 2 7 0x y- + = in surd form with rational denominator.
6. Prove that the line between ,1 4-^ h and ,3 3^ h is perpendicular to the line 4 6 0.x y- - =
7. Find the x - and y -intercepts of 2 5 10 0.x y- - =
8. (a) Find the equation of the straight line l that is perpendicular to the line
21 3y x= - and passes through , .1 1-^ h
(b) Find the x -intercept of l . (c) Find the exact distance from ,1 1-^ h
to the x -intercept of l .
9. Prove that lines 5 7y x= - and 10 2 1 0x y- + = are parallel.
10. Find the equation of the straight line passing through the origin and parallel to the line with equation 3 4 5 0.x y- + =
11. Find the point of intersection between lines 2 3y x= + and 5 6 0.x y- + =
12. The midpoint of ,a 3^ h and , b4-^ h is , .1 2^ h Find the values of a and b .
13. Find the acute angle between the lines 2 5 1 0x y- + = and 7 0x y+ - = to the nearest minute.
14. Show that the lines 4 0,x y- - =
,x y x y2 1 0 5 3 14 0+ + = - - = and 3 2 9 0x y- - = are concurrent.
15. Divide the interval between points ,3 4-^ h and ,2 2^ h in the ratio : .4 5
16. A straight line makes an angle of 153 29c l with the x -axis in the positive direction. What is its gradient, to 3 signifi cant fi gures?
17. The perpendicular distance from ,3 2-^ h to the line 5 12 0x y c- + = is 2. Find 2 possible values of c .
18. Find the equation of the straight line through ,1 3^ h that passes through the intersection of the lines 2 5 0x y- + = and 2 5 0.x y+ - =
19. Divide the interval between ,0 5^ h and ,2 4-^ h in the external ratio of : .2 3
20. The gradient of the line through ,3 4-^ h and ,x 2^ h is −5. Evaluate x .
21. Find the obtuse angle between the lines 3 3 0x y- + = and .x y2 5 1 0+ - =
ch7.indd 434 5/23/09 1:53:38 PM
435Chapter 7 Linear Functions
22. Show that the points ,2 1-^ h and ,6 3^ h are on opposite sides of the line .x y2 3 1 0- - =
23. Find the acute angle between the lines 3 4y x= - and .y x5= -
24. Find the equation of the line with x -intercept 4 that makes an angle of 45c with the x -axis.
25. Find the equation of the line with y -intercept 2- and perpendicular to the line passing through ,3 2-^ h and , .0 5^ h
1. If points , ,,k k k3 1 1 3- - -^ ^h h and ,k k4 5- -^ h are collinear, fi nd the value of k .
2. Find the equation, in exact form, of the line passing through ,3 2-_ i that makes an angle of 30c with the positive x -axis.
3. Find the equation of the circle whose centre is at the origin and with tangent 3 9 0.x y- + =
4. ABCD is a rhombus where , , , ,,A B C3 0 0 4 5 4= - = =^ ^ ^h h h and , .D 2 0= ^ h Prove that the diagonals are perpendicular bisectors of one another.
5. Prove that the points , ,1 2 2-_ i ,3 6-_ i and ,5 2-_ i all lie on a circle with centre the origin. What are the radius and equation of the circle?
6. Find the exact distance between the parallel lines 3 2 5 0x y+ - = and 3 2 1.x y+ =
7. A straight line has x -intercept ,aA 0^ h and y -intercept , ,B b0^ h where a and b are positive integers. The gradient of line AB is .1- Find OBA+ where O is the origin and hence prove that .a b=
8. Find the exact perpendicular distance between the line 2 3 1 0x y+ + =
and the point of intersection of lines 3 7 15x y- = and 4 5.x y- = -
9. Find the magnitude of the angle, in degrees and minutes, that the line joining ,1 3-^ h and ,2 4-^ h makes with the x -axis in the positive direction.
10. Find the equation of the line that passes through the point of intersection of lines 2 5 19 0x y+ + = and 4 3 1 0x y- - = that is perpendicular to the line 3 2 1 0.x y- + =
11. Prove , ,,A B2 5 4 5-^ ^h h and ,C 1 2-^ h are the vertices of a right-angled isosceles triangle.
12. Find the coordinates of the centre of a circle that passes through points , ,,7 2 2 3^ ^h h and , .4 1- -^ h
13. If 2 0ax y- - = and 5 11 0bx y- + = intersect at the point , ,3 4^ h fi nd the values of a and b .
14. Find the equation of the straight line through ,3 4-^ h that is perpendicular to the line with x -intercept and y -intercept −2 and 5 respectively.
15. Find the acute angle between the straight lines with equations 3 5x y- = and 2 4 1 0.x y- + =
Challenge Exercise 7
ch7.indd 435 6/26/09 4:14:33 AM
436 Maths In Focus Mathematics Extension 1 Preliminary Course
16. Find the exact equation of the straight line through the midpoint of , ,0 5-^ h and ,4 1-^ h that is perpendicular to the line that makes an angle of 30c with the x -axis.
17. Point ,x yP ^ h moves so that it is equidistant from points ,A 1 4^ h and , .B 2 7-^ h By fi nding the distances AP and BP , fi nd the equation of the locus of P .
18. Find the value of b if the lines 2 1 0x y- + = and 7 5 0bx y- + = make an angle of 45c at their intersection.
19. Find the coordinates of trisection of the interval between ,3 1-^ h and , .1 5-^ h
20. Prove that if two lines with gradients m1 and m2 meet at an angle of ,45c then 1m m m m1 2 1 2= - - or 1.m m m m1 2 2 1= - -
21. A and B have coordinates ,1 3^ h and ,4 7-^ h respectively. If P divides AB in the external ratio of : ,p 1 fi nd the coordinates of P in terms of p .
22. (a) Show that the point ,7 7-^ h lies on the line joining ,A 2 0-^ h and , .B 3 7-^ h
Find the ratio in which the point (b) divides AB .
23. The interval AB where ,A 5 3= -^ h and ,B x y= ^ h is divided by point P in the ratio of : .3 2 If the point P has coordinates , ,8 9-^ h fi nd values for x and y .
24. The angle between straight lines 2 3 0x y- = and 4 9mx y+ = is .32 51c l Find the value of m correct to 2 signifi cant fi gures.
25. Given points , ,,A B1 0 2 5^ ^h h and ,C 9 0^ h are the vertices of a triangle,
fi nd the coordinates of (a) P that divide AB in the ratio :2 1
fi nd the coordinates of (b) Q that divide CB in the ratio :2 1
prove (c) PQ AC< fi nd the coordinates of (d) R that divide
