Coordinate of straight lines.pdf

7/28/2019 Coordinate of straight lines.pdf

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Coordinate geometry is the use of algebraic methods to study

the geometry of straight lines and curves. In this chapter youwill consider only straight lines. Curves will be studied later.

3.1 Cartesian coordinatesIn your GCSE course you plotted points in two dimensions. This

section revises that work and builds up the terminology

required.

C H A P T E R 3

C1: Coordinate geometryof straight lines

Learning objectivesAfter studying this chapter, you should be able to:

use the language of coordinate geometry

find the distance between two given points

find the coordinates of the mid-point of a line segment joining two given points

find, use and interpret the gradient of a line segment know the relationship between the gradients for parallel and for perpendicular lines

find the equations of straight lines given (a) the gradient and y-intercept, (b) the gradient and

a point, and (c) two points

verify, given their coordinates, that points lie on a line

find the coordinates of a point of intersection of two lines

find the fourth vertex of a parallelogram given the other three.

positivey-axisy

xO

negativey-axis

negativex-axis

2nd quadrant 1st quadrant

3rd quadrant 4th quadrant

positivex-axis

The vertical line Oy is

called the y-axis.

The horizontal line Oxiscalled the x-axis.

The point O, where the

coordinate axes cross,is called the origin.

Named after Ren Descartes(15961650), a French

mathematician.

The plane is divided up into four

quadrants by extending two

perpendicular lines called the

coordinate axes Oxand Oy.


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C1: Coordinate geometry of straight l ines 29

In the diagram below, the pointP lies in the first quadrant.

Its distance from they-axis isa.

Its distance from thex-axis is b.

The diagram below shows part of the Cartesian grid, where L is

the point (2, 3).

y

c

a P

Q

R

S (g,h)

b

h

g

e

f

d

xO

Q is in the second quadrant.Thex-coordinate is negative andthey-coordinate is positive.

The pointP has Cartesiancoordinates (a, b)

Coordinates ofQare (c,d)

R is the point(e, f)

The originO is (0, 0)

We say that:

the x-coordinate ofP is athe y-coordinate ofP is b.

We write P is the point (a, b) or

simply the point P(a, b).

yL (2, 3)

M (0, 2)

J(1, 0) K(3, 0)

N(0, 3)

xO

Any point on they-axis has itsx-coordinateequal to 0.

Any point on thex-axis has itsy-coordinate equalto 0. JandKarepoints on thex-axis.

3

2

1

1

2

3

34 2 1 1 2 3 4

The x-coordinate ofL is 2 andthe y-coordinate ofL is 3.


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0 C1: Coordinate geometry of straight l ines

Worked example 3.11 Draw coordinate axes Oxand Oy and plot the points

A(1, 2), B(3, 3), C(1,4) andD(2,4).

2 The line joining the points CandD crosses they-axis at thepointE. Write down the coordinates ofE.

3 The line joining the pointsA and Ccrosses thex-axis at thepointF. Write down the coordinates ofF.

Solution1

2 E is on they-axis so itsx-coordinate is 0. E(0,

4).3 Fis on thex-axis so itsy-coordinate is 0. F(1, 0).

3.2 The distance between two pointsThe following example shows how you can use Pythagoras

theorem which you studied as part of your GCSE course, in

order to find the distance between two points.

Worked example 3.2Find the distanceABwhereA is the point (1, 2) andB is thepoint (4, 4).

SolutionFirst plot the points, join them with a line and make a

right-angled triangleABC.

The distanceAC 4 1 3.

The distanceBC 4 2 2.

Using Pythagoras theorem AB2 32 22. 9 4 13

AB 13

Ox

y

B(3, 3)

A(1, 2)

C(1,4)

D(2, 4)

4321 1 2 3 4 5

1

5

4

3

2

1

2

3

4

5

From the diagram in Workedexample 3.1, you can see that the

distance between the pointsC(1, 4) and D(2,4) is 3

units and the distance betweenthe points A(1, 2) and C(1, 4) is6 units, but how do you find the

distance between points whichdo not lie on the same horizontal

line or on the same vertical line?

AB 3.6 (to two significantfigures) but since a calculator isnot allowed in the examination

for this unit you should leave

your answers in exact forms.Where possible surd answers

should be simplified.

O x

y

CA

B

1 2 3 4

1

2

3

4


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We can generalise the method of the previous example to find a

formula for the distance between the pointsP(x1,y1) and

Q(x2,y2).

The distancePRx2x1.

The distance QRy2y1.

Using Pythagoras theorem

PQ2 (x2x1)2 (y2y1)

2

PQ [(x2x1)2(y2y1)

2].

Worked example 3.3The pointR has coordinates (1, 2) and the pointS has

coordinates (5, 6).

(a) Find the distanceRS.

(b) The point Thas coordinates (0, 9). Show thatRThas lengthk 2, wherek is an integer.

Solution(a) For pointsR andS:

the difference between thex-coordinates is 5 (1) 6

the difference between they-coordinates is 6 28.

RS (6)2(8)2

36 64 100 10

(b) RT (1)2(7)2 1 49 50

25 2

25 2

5 2

Worked example 3.4The distanceMNis 5, whereM is the point (4,2) andNis the

point (a, 2a). Find the two possible values of the constanta.


O x1

1

2

x2

y

x

P(x1, y1)

Q(x2, y2)

R

The distance between the points (x1,y1) and (x2,y2) is

(x2x1)2(y2y1)

2.

An integer is a whole number.

If you draw a diagram you willprobably find yourself dealing

with distances rather than thedifference in coordinates inorder to avoid negative numbers.

This is not wrong.

Of course you could have

considered the differences as1 56 and 2 (6) 8

and obtained the same answer.

Used a b a b. Seesection 2.4.

You may prefer to draw adiagram but it is also good tolearn to work algebraically.


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SolutionThe difference between thex-coordinates isa 4.

The difference between they-coordinates is 2a (2) 2a 2.

MN2 (a 4)2 (2a 2)2

52 (a 4)2 (2a 2)2

25 a2 8a 16 4a2 8a 4

25 5a2 20

a2 1

a1.

EXERCISE 3A

1 Find the lengths of the line segments joining:

(a) (0, 0) and (3, 4), (b) (1, 2) and (5, 3),(c) (0, 4) and (5, 1), (d) (3, 1) and (1, 6),

(e) (4,2) and (3, 0), (f) (3,2) and (6, 1),

(g) (2, 7) and (3,1), (h) (2, 0) and (6,3),

(i) (1.5, 0) and (3.5, 0) (j) (2.5, 4) and (1, 6),

(k) (8, 0) and (2, 2.5), (l) (3.5, 2) and (4, 8).

2 Calculate the lengths of the sides of the triangle ABCand

hence determine whether or not the triangle is right-angled:

(a) A(0, 0) B(0, 6) C(4, 3),

(b) A(3, 0) B(1, 8) C(7, 6),

(c) A(1, 2) B(3, 4) C(0, 7).

3 The vertices of a triangle areA(1, 5),B(0, 2) and C(4, 2). Bywriting each of the lengths of the sides as a multiple of 2,show that the sum of the lengths of two of the sides is three

times the length of the third side.

4 The distance between the two pointsA(6, 2p) andB(p,3) is5 5. Find the possible values ofp.

5 The vertices of a triangle areP(1, 3), Q(2, 0) andR(4, 0).

(a) Find the lengths of the sides of trianglePQR.

(b) Show that angle QPR 90.

(c) The line of symmetry of trianglePQR meets thex-axis at

pointS. Write down the coordinates ofS.(d) The point Tis such thatPQTR is a square. Find the

coordinates ofT.


Using (xy)2x2 2xyy2.

Notice that this equation doeshave two solutions.

The identity (3k)2 (4k)2 (5k)2

may be useful in (j), (k) and (l).

From question 2 onwards you

are advised to draw a diagram.

In longer questions, the resultsfound in one part often help in

the next part.


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3.3 The coordinates of the mid-pointof a line segment joining twoknown points

From the diagram you can see that the mid-point of the

line segment joining (0, 0) to (6, 0) is (3, 0) or

or 026, 0,

the mid-point of the line segment joining (0, 0) to (0,4)

is (0,2) or

0,02(4),

and the mid-point of the line segment joining (0, 4) to (6, 0)

is (3,2) or

026,

(4

2

) 0.

Going fromP(1, 2) to Q(5, 8) you move 4 units horizontally and

then 6 vertically.

IfM is the mid-point ofPQ then the journey is halved so to go

fromP(1, 2) toMyou move 2 (half of 4) horizontally and then

3 (half of 6) vertically.

SoM is the point (1 2, 2 3) orM(3, 5) orM

1

2

5,

2

2

8

.

Note thatx1 12(x2x1)

12(x1x2) which is thex-coordinate ofM.


y

x

mid-point ofline segmentis (3, 2)

mid-point

mid-point

1 2 4 5 6 7

2

1

O1

2

3

4

5

3

x

y

2 along

6 up

Q(5, 8)

M

3 up

1 2 3 4 5 6O

4

5

6

7

8

9

3

2

14 along

P(1, 2)

x

y

O

A(x1, y1)

B(x2, y2)

(x2x1)

(y2 y1) up

M(y2 y1)x1 x2

2

y1 y22 ,

12

(x2 x1) along12

In general, the coordinates of the mid-point of the

line segment joining (x1,y1) and (x2,y2) are

x12x2,

y1

2

y2.

Check that M to Q is also 2along and then 3 up.


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Worked example 3.5M is the mid-point of the line segment joiningA(1,2) and

B(3, 5).

(a) Find the coordinates ofM.

(b) M is also the mid-point of the line segment CDwhereC(1, 4). Find the coordinates ofD.

Solution

(a) Using x12x2,

y1

2

y2,M is the point 12

3,

2

2

5

soM(2, 112).

(b) LetD be the point (a, b) then the mid-point ofCD is

a

2

1,

b

2

4

(2, 1

1

2

)

For this to be truea

2

1 2 and

b

2

4

3

2a 3 and

b1.

The coordinates ofD are (3,1).

EXERCISE 3B

1 Find the coordinates of the mid-point of the line segmentsjoining:

(a) (3, 2) and (7, 2), (b) (1,2) and (1, 3),

(c) (0, 3) and (6, 1), (d) (3, 3) and (1, 6),

(e) (4,2) and (3, 6), (f) (3,2) and (6, 1),

(g) (2, 5) and (2,1), (h) (2, 5) and (6,3),

(i) (1.5, 6) and (3.5, 0) (j) (3.5, 2) and (4,1).

2 M is the mid-point of the straight line segment PQ. Find thecoordinates ofQ for each of the cases:

(a) P(2, 2),M(3, 4), (b) P(2, 1),M(3, 3),(c) P(2, 3),M(1, 5), (d) P(2,5),M(3, 0),

(e) P(2, 4),M(1, 212), (f) P(1,3),M(2

12,4

12).

3 The mid-point ofAB, whereA(3,1) andB(4, 5), is also themid-point ofCD, where C(0, 1).

(a) Find the coordinates ofD.

(b) Show thatACBD.

4 The mid-point of the line segment joining A(1, 3) andB(5,1) isD. The point Chas coordinates (4, 4). Show that

CD is perpendicular toAB.


x

y

M(2, 1 )

B(3, 5)C(1, 4)

A(1,2)

D

1 2 3 4 5O

1

1

2

2

3

4

5

12

Or, using the idea of journeys:

Cto M is 1 across and 212 down.

So M to D is also 1 across and 212

down giving D(2 1, 112 212) or

D(3,1).

coordinates ofM


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3.4 The gradient of a straight linejoining two known points

The gradient of a straight line is a measure of how steep it is.

The gradient of the line joining two points .

Lines which are equally steep are parallel.

Parallel lines have equal gradients.

x

y

B(x2, y2)

A(x1, y1)

(y2y1)

O

(x2x1)

Change inx

Change iny

change iny-coordinate

change inx-coordinate


x

y

(1)

(2)

(3)

O x

y

O x

y

O

These three lines slope upwards fromleft to right. They have gradients

which are positive. Line (2) is steeperthan line (1) so the gradient of (2)is greater than the gradient of (1).

These three lines are all parallel tothex-axis. They are not sloping.Horizontal lines have gradient 0.

These three lines slope downwardsfrom left to right. They havegradients which are negative.

(a) (b) (c)

The gradient of the line joining the two points

A(x1,y1) andB(x2,y2) is

x

y2

2

y

x

1

1

.


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Worked example 3.6O(0, 0),P(3, 6), Q(0, 5) andR(2, 1) are four points.

(a) Find the gradient of the line segment (i) OP, (ii) RQ.

(b) Find the gradient of the line segment (i) OR, (ii) PQ.

(c) What can you deduce from your answers?

Solution

(a) (i) Gradient ofOP3

6

0

0 2;

(ii) Gradient ofRQ

0

5

(

1

2)

2.

(b) (i) Gradient ofOR

1

2

0

0

1

2;

(ii) Gradient ofPQ6

3

5

0

1

3.

(c) The lines OP andRQ have gradients which are equal so they

are parallel. Lines OR andPQ are not parallel since their

gradients are not equal.

So we can deduce that the quadrilateral OPQR is a trapezium.

EXERCISE 3C

1 By finding the gradients of the linesAB and CD determine ifthe lines are parallel.

(a) A(2, 3) B(3, 5) C(0, 1) D(1, 3),

(b) A(3, 2) B(5, 1) C(4,3) D(2,2),

(c) A(4, 5) B(4, 5) C(1,2) D(0,2),

(d) A(6,3) B(1,2) C(312, 0) D(7,

12).

2 By finding the gradients of the linesAB andBCshow thatA(2, 3),B(2, 2) and C(6, 1) are collinear points.

3 A(1,3),B(4,2) and C(612, 0) are the vertices of triangleABC.

(a) Find the gradient of each side of the triangle.(b) Which side of the triangle is parallel to OPwhere O is the

origin andP is the point (11,6)?


x

y

P(3, 6)

Q(0, 5)

R(2, 1)

O

A diagram is very helpful. You

should try to position the pointsroughly in the correct placewithout plotting the points on

graph paper, then you might seeproperties such as parallel lines

and possible right angles.

Collinear means in a straightline.


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3.5 The gradients of perpendicularlines

Rotate the shaded triangles clockwise through 90 as shown,

keeping O fixed.

Line OA line OA Line OB line OB

A(3, 1)A(1,3) B(4, 2) B(2, 4)

Gradient ofOA 1

3 Gradient ofOB

2

4

Gradient ofOA

1

3 Gradient ofOB

4

2

OA is perpendicular to OA

OB is perpendicular to OB

Gradient ofOAGradient OA Gradient ofOBGradient OB

1

3

1

31

2

4

4

21

In general

gradient ofOP b

a gradient ofOP

a

b

Lines are perpendicular if the product of their gradientsis 1.

Worked example 3.7Find the gradient of a line which is perpendicular to the line

joiningA(1, 3) andB(4, 5).

x

y

A(3, 1)

O

x

y

(a) (b)

O

1

1

2

3

2

3

1

2

3

4

5

12

12345

1

4321

2 3

A(1, 3)

B(2, 4)

B(4, 2)


Lines with gradients m1 and m2

are parallel ifm1m2,

are perpendicular ifm1m21.

x

y

P(a, b)

P(b, a)

O


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Solution

Gradient ofAB5

4

3

1

2

3.

Let m2 be the gradient of any line perpendicular toAB then

2

3m21 m2

3

2.

The gradient of any line perpendicular toAB is 3

2.

EXERCISE 3D

1 Write down the gradient of lines perpendicular to a line withgradient:

(a) 2

5

, (b)

3

1, (c) 4, (d) 3

12, (e) 2

14.

2 Two vertices of a rectangleABCD areA(2, 3) andB(4, 1). Find:

(a) the gradient ofDC,

(b) the gradient ofBC.

3 A(1, 2),B(3, 6) and C(7, 4) are the three vertices of a triangle.

(a) Show thatABCis a right-angled isosceles triangle.

(b) D is the point (5, 0). Show thatBD is perpendiculartoAC.

(c) Explain whyABCD is a square.

(d) Find the area ofABCD.

3.6 The y= mx+ c form of theequation of a straight line

Consider any straight line which crosses the y-axis at the point

A(0, c). We say that c is the y-intercept.LetP(x,y) be any other point on the line.

If the gradient of the line is m then

x

y

0

cm

soy cmx or ymx c.


ymx c is the equation of a straight line with gradient

m andy-intercept c.

ax

by

c

0 is the general equation of a line. It hasgradient

a

b andy-intercept

b

c.

x

y

A(0, c)

yc

x 0

P(x,y)

O

1

m


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3.7 The y y1 = m(x x1) form of theequation of a straight line

Consider any line which passes through the known point

A(x1,y1) and letP(x,y) be any other point on the line.

Ifm is the gradient of the lineAP, thenxy

yx

1

1m

oryy1m(xx1).

Worked example 3.10Find the equation of the straight line which is parallel to the

liney 4x 1 and passes through the point (3, 2).

SolutionThe gradient of the liney 4x 1 is 4,

so the gradient of any line parallel toy 4x 1 is also 4.

We need the line with gradient 4 and through the point (3, 2),

so its equation is, using yy1m(xx1),

y 2 4(x 3)

ory 4x 10.

Worked example 3.11(a) Find a Cartesian equation for the perpendicular bisector of

the line joiningA(2, 1) andB(4,5).

(b) This perpendicular bisector cuts the coordinate axes at CandD. Show that CD 1.5AB.

x

y

D

C

O

5

2 4

A(2, 1)

B(4, 5)


The equation of the straight line which passes through the

point (x1,y1) and has gradient m is

yy1m(xx1).

Compare y 4x 1

with ymx c

Parallel lines have equal gradients.

x

y

yy1

P(x,y)

O

A(x1, y1)

xx1


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SolutionFirst, we draw a rough sketch.

(a) The gradient ofAB

4

5

2

13,

so the gradient of the perpendicular is 1

3.

The mid-point ofAB is 224,

1

2

(5) (3,2).

The perpendicular bisector is a straight line which passes

through the point (3,2) and has gradient 1

3, so its

equation is

y (2) 1

3(x 3) or y

1

3x 3.

(b) Let Cbe the point where the liney 1

3x 3 cuts they-axis.

Whenx 0,y 1

3 0 33, so we have C(0,3).

LetD be the point where the liney 1

3x 3 cuts thex-axis.

Wheny 0, 0 1

3 x 3

x

9, so we haveD(9, 0)

Distance CD (90)2(0(3))2 819 90

DistanceAB (42)2(51)2 436 40

C

A

D

B

9

4

0

094

0

094

9

4

3

2 1.5

so CD 1.5AB.

EXERCISE 3F

1 Find an equation for the straight line with gradient 2 andwhich passes through the point (1, 6).

2 Find a Cartesian equation for the straight line which has

gradient 1

3 and which passes through (6, 0).

3 Find an equation of the straight line passing through (1, 2)which is parallel to the line with equation 2yx4.

4 Find an equation of the straight line that is parallel to3x 2y 4 0 and which passes through (1, 3).


m1m21

Using x12x2,

y1

2

y2.

Using yy1m(xx1).

Any point on the y-axis has

x-coordinate 0.

Any point on the x-axis has

y-coordinate 0.

Using (x2x1)2(y2y1)

2.

Because a straight line equationcan be arranged in a variety of

ways you are usually asked tofind an equation rather than theequation.

All correct equivalents wouldscore full marks unless you have

been asked specifically for a

particular form of the equation.

Using ab .See section 2.4.

a

b


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5 Find an equation of the straight line which passes through

the origin and is perpendicular to the liney 1

2x 3.

6 Find they-intercept of the straight line which passesthrough the point (2, 2) and is perpendicular to the line

3y

2x

1.

7 Find a Cartesian equation for the perpendicular bisector ofthe line joiningA(2, 3) andB(0, 6).

8 The vertex,A, of a rectangleABCD, has coordinates (2, 1). The

equation ofBCisy1

2x 3. Find, in the formymx c,

the equation of:

(a) AD,

(b) AB.

9 Given the pointsA(0, 3),B(5, 4), C(4,1) andE(2, 1):

(a) show thatBE is the perpendicular bisector ofAC,

(b) find the coordinates of the pointD so thatABCD is arhombus,

(c) find an equation for the straight line throughD andA.

10 The perpendicular bisector of the line joiningA(0, 1) and

C(4,

7) intersects thex-axis atB and they-axis atD.Find the area of the quadrilateralABCD.

11 Show that the equation of any line parallel to ax by c 0is of the formax byk 0.

3.8 The equation of a straight linepassing through two given points

Worked example 3.12Find a Cartesian equation of the straight line which passes

through the points (2, 3) and (1, 0).


Hint for (b). The diagonals of a

rhombus are perpendicular andbisect each other.

The equation of the straight line which passes through thepoints (x1,y1) and (x2,y2) is

y

y

2

y

y

1

1

x

x

2

x

x

1

1

The derivation of this equation is similar to previous ones and is left as anexercise to the reader.

x

y P(x,y)

O

A(x1, y1)

B(x2, y2)


16/24

SolutionIf you take (x1,y1) (2, 3) and (x2,y2) = (1, 0) and

substitute into the general equation

y

y

2

y

y

1

1

x

x

2

x

x

1

1

,

you get

0

y

3

3

x

1

2

2,

or y 3x 2 which leads toyx 1.

3.9 The coordinates of the point of

intersection of two linesIn this section you will need to solve simultaneous equations. If

you need further practice, see section 1.3 of chapter 3.

When two lines intersect, the point of intersection lies on both

lines. The coordinates of the point of intersection must satisfythe equations of both lines. The equations of the lines must be

satisfied simultaneously.

From the diagram, you can clearly read off (1, 2) as the point ofintersection of the linesxy 1 0 and 2xy 4 0. So the

solution to the simultaneous equationsxy 1 0 and

2xy 4 0 isx 1,y 2.


Note that you can check that yourline passes through the points (2, 3)

and (1, 0) by seeing if the pointssatisfy the equation yx 1.

yx 1

Checking for (2, 3)LHS 3 RHS 2 1 3

Checking for (1, 0)

LHS 0 RHS1 1 0

A point lies on a line if the coordinates of the point satisfy

the equation of the line.

1 20

1

2

3

x

4

2xy 4 0

xy 1 0

(1, 2)

Given accurately drawn graphs of the two intersecting

straight lines with equationsax by c 0 and

AxBy C 0, the coordinates of the point of intersection

can be read off. These coordinates give the solution of the

simultaneous equationsax by c 0 andAxBy C 0.

To find the coordinates of the point of intersection of

the two lines with equationsax by c 0 and

AxBy C 0, you solve the two linear equations

simultaneously.

(x,y)


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Worked example 3.13Find the coordinates of the point of intersection of the straight

lines with equationsyx 2 andy 4x 1.

SolutionAt the point of intersectionP,yx 2 andy 4x 1.

So eliminatingy gives 4x 1x 2.

Rearranging gives 4xx 2 1

3x 3x 1.

Thex-coordinate ofP is 1.

To find they-coordinate ofP, putx 1 intoyx 2 y 3

The point of intersection is (1, 3).

Worked example 3.14

The straight lines with equations 5x

3y

7 and 3x

7y

13intersect at the pointR. Find the coordinates ofR.

SolutionTo find the point of intersectionR, you need to solve the

simultaneous equations

5x 3y 7 [A]

3x 7y 13 [B]

35x 21y 49 [C]

9x 21y 39 [D]

44x 0 88

x 2.

Substitutingx 2 in [A] gives 10 3y 7

3y3 y1.

The coordinates of the pointR are (2,1).

x

y

P

O1

2


Checking that (1, 3) lies on theline yx 2;

LHS 3 RHS 1 2 3

Checking that (1, 3) lies on the

line y 4x 1;

LHS 3 RHS 4 1 3

Hint. Checking that the coordinates satisfy each line equation is advisable,

especially if the result is being used in later parts of an examinationquestion.

It can usually be done in your head rather than on paper.

Multiply equation [A] by 7 and

equation [B] by 3.

Adding [C] [D].

Checking in [A]

LHS10(3)7RHS ;

and in [B]

LHS6 7(1) 13

x

y

R

O


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EXERCISE 3G

1 Verify that (2, 5) lies on the line with equationy 3x 1.

2 Which of the following points lie on the line with equation3x 2y 6:

(a) (3, 0), (b) (2, 0), (c) (4, 3)

(d) (2, 6), (e) (0, 2)?

3 Find the coordinates of the points where the following linesintersect thex-axis:

(a) yx 4, (b) y 2x 6,

(c) 2x 3y 6 0, (d) 3x 4y 12 0.

4 The point (k, 2k) lies on the line with equation 2x 3y 6 0.Find the value ofk.

5 Show that the point (4, 8) lies on the line passing throughthe points (1, 3) and (7,3).

6 (a) Find the equation of the lineABwhereA is the point(3, 7) andB is the point (5,1).

(b) The point (k, 3) lies on the lineAB. Find the value of theconstantk.

7 A(5, 2),B(2, 3), C(2,1) andD(4,2) are the verticesof the quadrilateralABCD.

(a) Find the equation of the diagonalBD.

(b) Determine whether or not the mid-point ofAClies on thediagonalBD.

8 Find the coordinates of the point of intersection of these pairsof straight lines:

(a) y 2x 7 andyx 1,

(b) 3yx 7 and 2yx 3,

(c) 5x 2y 16 and 3x 2y 8,

(d) y 8xandy 40 3x,

(e) y7 and 5yx 1,(f) y 3x 3 and 2y 5x 9,

(g) 4y 9x 8 and 5y 6x 3,

(h) 8y 3x 11 and 2x 5y 6.

9 PointA has coordinates 121, 1, pointB has coordinates

3, 661

0 and point Chas coordinates 16

9,

3

5. The straight

lineAChas equation 12x 65y 1 0 and the straight line

BChas equation 60y 150x 511. Write down the solution ofthe simultaneous equations 12x 65y 1 0 and

60y 150x 511.



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20/24


21/24

2 The lineAB has equation 5x 2y 7. The pointA hascoordinates (1, 1) and the pointB has coordinates (3,k).

(a) (i) Find the value ofk.

(ii) Find the gradient ofAB.

(b) Find an equation for the line through Awhich is

perpendicular toAB.(c) The point Chas coordinates (6, 2). Show thatAChas

lengthp 2, stating the value ofp. [A]

3 The pointP has coordinates (1, 10) and the point Q hascoordinates (4, 4).

(a) Show that the length ofPQ is 3 5.

(b) (i) Find the equation of the perpendicular bisector ofPQ.

(ii) This perpendicular bisector intersects thex-axis atthe pointA. Find the coordinates ofA.

4 The pointA has coordinates (3, 5) and the pointB hascoordinates (1, 1).

(a) (i) Find the gradient ofAB.

(ii) Show that the equation of the line AB can be

written in the form rxys, where rands are

positive integers.

(b) The mid-point ofAB isM and the lineMCisperpendicular toAB.

(i) Find the coordinates ofM.

(ii) Find the gradient of the line MC.(iii) Given that Chas coordinates (5,p), find the value

of the constantp. [A]

5 The pointsA andB have coordinates (13, 5) and (9, 2),respectively.

(a) (i) Find the gradient ofAB.

(ii) Find an equation for the lineAB.

(b) The point Chas coordinates (2, 3) and the pointXliesonAB so thatXCis perpendicular toAB.

(i) Show that the equation of the line XCcan be

written in the form 4x 3y 17.(ii) Calculate the coordinates ofX. [A]

6 The equation of the lineAB is 5x 3y 26.

(a) Find the gradient ofAB.

(b) The pointA has coordinates (4, 2) and a point Chascoordinates (6, 4).

(i) Prove thatACis perpendicular toAB.

(ii) Find an equation for the lineAC, expressing youranswer in the formpxqy r, wherep,q and rare

integers.(c) The line with equationx 2y 13 also passes through

the pointB. Find the coordinates ofB. [A]



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7 The pointsA,B and Chave coordinates (1, 7), (5, 5) and(7, 9), respectively.

(a) Show thatAB andBCare perpendicular.

(b) Find an equation for the lineBC.

(c) The equation of the lineACis 3yx 20 andM is themid-point ofAB.

(i) Find an equation of the line through M parallel toAC.

(ii) This line intersectsBCat the point T. Find thecoordinates ofT. [A]

8 The pointA has coordinates (3, 5),B is the point (5, 1)and O is the origin.

(a) Find, in the formymx c, the equation of the

perpendicular bisector of the line segmentAB.

(b) This perpendicular bisector cuts they-axis atP and the

x-axis at Q.

(i) Show that the line segmentBP is parallel to thex-axis.

(ii) Find the area of triangle OPQ.

9 The pointsA(1, 2) and C(5, 1) are opposite vertices of a

parallelogramABCD. The vertexB lies on the line 2xy 5.

The sideAB is parallel to the line 3x 4y 8. Find:

(a) the equation of the sideAB,

(b) the coordinates ofB,

(c) the equations of the sidesAD and CD,

(d) the coordinates ofD.

10 ABCD is a rectangle in which the coordinates ofA and Care(0, 4) and (11, 1), respectively, and the gradient of the side

AB is 5.

(a) Find the equations of the sidesAB andBC.

(b) Show that the coordinates ofB is (1, 1).

(c) Calculate the area of the rectangle.

(d) Find the coordinates of the point on they-axis which isequidistant from pointsA andD. [A]



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1 The distance between the points (x1,y1) and (x2,y2) p31

is (x2x1)2(y2y1)

2.

2 The coordinates of the mid-point of the line segment p33

joining (x1,y1) and (x2,y2) are x1

2x2

,

y12

y2.

3 The gradient of a line joining the two pointsA(x1,y1) p35

andB(x2,y2)x

y2

2

y

x

1

1

.

4 Lines with gradients m1 and m2: p37 are parallel ifm1m2,

are perpendicular ifm1m21.

5 ymx c is the equation of a straight line with p38gradient m andy-intercept c.

6 ax by c 0 is the general equation of a line. p38

It has gradient a

b andy-intercept

b

c.

7 The equation of the straight line which passes p40

through the point (x1,y1) and has gradient m is

yy1m(xx1).

8 The equation of the straight line which passes p42

through the points (x1,y1) and (x2,y2) is

y

y

2

y

y

1

1

x

x

2

x

x

1

1

.

9 A point lies on a line if the coordinates of the point p43

satisfy the equation of the line.

10 Given accurately drawn graphs of the two intersecting p43

straight lines with equationsax by c 0 and

AxByC 0, the coordinates of the point of intersection

can be read off. These coordinates give the solution of the

simultaneous equationsax by c 0 andAxByC 0.

11 To find the coordinates of the point of intersection of p43

the two lines with equationsax by c 0 and

AxBy C 0, you solve the two equations

simultaneously.

Key point summary


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1 Calculate the distance between the points (2,3) and (7, 9). Section 3.2

2 State the coordinates of the mid-point of the line segment PQ Section 3.3whereP(3,2) and Q(7, 1).

3 Find the gradient of the line joining the pointsA andB Section 3.4whereA is the point (3,2) andB is the point (5, 4).

4 The lines CD andEFare perpendicular with points C(1, 2), Section 3.5D(3,4),E(2, 5) andF(k, 4). Find the value of the constantk.

5 Find a Cartesian equation of the line which passes through Sections 3.5 and 3.7the point (2, 1) and is perpendicular to the line 5y 3x 7.

6 Find the point of intersection of the lines with equations Section 3.93x 5y 11 andy 4x 9.

What to reviewTest yourself

113.

2(5,12).

33.

4k5.

53y5x13.

6(2,1).

ANSWERSTest yourself

Documents

Coordinate of straight lines.pdf