maths in focus - margaret grove - ch7

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 352 7/18/09 3:14:41 PM

353Chapter 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.

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354 Maths In Focus Mathematics 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 354 7/18/09 11:22:10 AM


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 355 7/18/09 11:22:15 AM


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 356 7/18/09 11:22:17 AM


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 357 7/31/09 4:47:03 PM


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 358 7/18/09 11:22:20 AM


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 359 7/18/09 11:22:21 AM


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 360 7/31/09 4:47:04 PM


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 361 7/18/09 11:22:22 AM


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 362 7/18/09 11:22:23 AM


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 363 7/18/09 11:22:24 AM


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 364 7/18/09 11:22:26 AM


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 365 7/18/09 11:22:26 AM


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 366 7/31/09 4:47:04 PM


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.

(a) y x= (b) 1y x= + (c) 2y x= + (d) 2y x= - (e) 3y x= -

Can you fi nd a pattern for the y -intercept of each line? Can you predict what the y -intercept of 11y x= + and 6y x= - would be?

hasy mx b= + gradientm =

b y= -intercept

ch7.indd 367 7/18/09 11:22:31 AM


EXAMPLES

1. Find the gradient and y -intercept of the linear function 7 5y x= - .

Solution

The equation is in the form y mx b= + where 7m = and 5b = - .

Gradient 7=

y-intercept 5= -

2. Find the gradient of the straight line with equation .x y2 3 6 0+ - =

Solution

First, we change the equation into the form y mx b= + .

So the gradient is .

x y

x y

x y

x y

y x

x

y x

y x

x

m

x x

2 3 6 0

2 3 6 0

2 3 6

2 3 6

3 6 2

2 6

3 2 6

32

36

32 2

32

32

6 6

2 2

3 3

+ - =

+ - =

+ =

+ =

= -

= - +

=- +

=-

+

= - +

= -

-

+ +

- -

There is a general formula for fi nding the gradient of a straight line.

The gradient of the line 0ax by c+ + = is given by

mba

= -

Proof

0ax by c

by ax c

ybax

bc

+ + =

= - -

= - -

mba

` = -

ch7.indd 368 7/18/09 11:22:31 AM


EXAMPLE

Find the gradient of 3 2x y- = .

Solution

,

x y

x y

a b

mba

3 2

3 2 0

3 1

13

3

3gradient is`

- =

- - =

= = -

= -

= --

=

7.4 Exercises

1. Find (i) the gradient and (ii) the y -intercept of each linear function.

(a) 3 5y x= + (b) 2 1f x x= +] g (c) 6 7y x= - (d) y x= - (e) 4 3y x= - + (f) 2y x= - (g) 6 2f x x= -] g (h) 1y x= - (i) 9y x= (j) 5 2y x= -

2. Find (i) the gradient and (ii) the y -intercept of each linear function .

(a) 2 3 0x y+ - = (b) 5 6 0x y+ + = (c) 6 1 0x y- - = (d) 4 0x y- + = (e) 4 2 1 0x y+ - = (f) 6 2 3 0x y- + = (g) 3 6 0x y+ + = (h) 4 5 10 0x y+ - = (i) 7 2 1 0x y- - = (j) 5 3 2 0x y- + =

3. Find the gradient of the straight line .

(a) 4y x= (b) 2 1y x= - - (c) 2y = (d) 2 5 0x y+ - = (e) 1 0x y+ + = (f) 3 8x y+ = (g) 2 5 0x y- + = (h) 4 12 0x y+ - = (i) 3 2 4 0x y- + = (j) 5 4 15x y- =

(k) 32 3y x= +

(l) 2

y x=

(m) 5

1y x= -

(n) 72 5y x

= +

(o) 53 2y x

= - -

(p) 27 3

1y x= - +

(q) 35

8xy

- =

(r) 2 3

1x y+ =

(s) 32 4 3 0x y- - =

(t) 4 3

27 0x y

+ + =

ch7.indd 369 7/18/09 11:22:32 AM


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 370 7/18/09 11:22:33 AM


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 371 7/18/09 11:22:33 AM


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 372 7/18/09 11:22:34 AM


( )

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 373 7/31/09 4:47:05 PM


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 374 7/18/09 11:22:36 AM


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 375 7/18/09 11:22:37 AM


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.

(a) 1. 3 4 12 0x y- + = (b) 4 3 8 0x y+ - = (a) 2. 2 4 0x y+ + = (b) 2 2 0x y- + =

What do you notice about these pairs of lines?

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 376 7/18/09 11:22:38 AM


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 377 7/18/09 11:22:39 AM


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 378 7/18/09 11:22:41 AM


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 379 7/18/09 11:22:42 AM


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 380 7/18/09 11:22:44 AM


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 381 7/18/09 11:22:45 AM


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 382 7/18/09 11:22:46 AM


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 383 7/18/09 11:22:47 AM


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 384 7/18/09 11:22:48 AM


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 385 7/18/09 11:22:50 AM


| |

.

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 386 7/18/09 11:22:52 AM


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 387 7/18/09 11:22:55 AM


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.

ch7.indd 388 7/18/09 11:22:56 AM


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. 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.

14. 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?

15. The perpendicular distance from ,3 2-^ h to the line 5 12 0x y c- + = is 2. Find 2 possible values of c .

16. 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+ - =

17. The gradient of the line through ,3 4-^ h and ,x 2^ h is −5. Evaluate x .

18. Show that the points ,2 1-^ h and ,6 3^ h are on opposite sides of the line .x y2 3 1 0- - =

19. Find the equation of the line with x -intercept 4 that makes an angle of 45c with the x -axis.

20. Find the equation of the line with y -intercept 2- and perpendicular to the line passing through ,3 2-^ h and , .0 5^ h

Test Yourself 7

ch7.indd 389 7/18/09 11:22:59 AM


Challenge Exercise 7

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 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.

16. 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 .

ch7.indd 390 7/18/09 11:23:05 AM

maths in focus - margaret grove - ch7

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