Cambridge International Mathematics

256 Coordinate geometry (Chapter 12)

The initial approaches used by these mathematicians were quite opposite. Descartes began with a line or

curve and then found the equation which described it. Fermat, to a large extent, started with an equation

and investigated the shape of the curve it described. This interaction between algebra and geometry

shows the power ofanalytical geometryas a branch of mathematics.

Analytical geometry and its use of coordinates provided the mathematical tools which enabled Isaac

Newton to later develop another important branch of mathematics called calculus.

Newton humbly stated: “If I have seen further than Descartes, it is because I have stood on the shoulders

of giants.”

THE NUMBER PLANE

The position of any point in thenumber planecan be specified in terms of anordered pair

of numbers (x,y), where:

xis thehorizontal stepfrom a fixed point ororiginO, and

yis thevertical stepfrom O.

Once the origin O has been given, two perpendicular axes are drawn.

Thex-axisis horizontal and they-axisis vertical.

Thenumber planeis also known as either:

² the2-dimensional plane,or

² theCartesian plane, named afterRene Descartes ́.

In the diagram, the point P is at(a,b).

aandbare referred to as thecoordinatesof P.

a is called thex-coordinate, and

b is called they-coordinate.

To plot the point A(3,4):

² start at the origin O

² move right along thex-axis 3 units

² then move upwards 4 units.

To plot the point B(5,¡2):

² start at the origin O

² move right along thex-axis 5 units

² then move downwards 2 units.

To plot the point C(¡ 4 ,1):

² start at the origin O

² move left along thex-axis 4 units

² then move upwards 1 unit.

A PLOTTING POINTS [7.1]

DEMO

DEMO

y-axis

O x-axis

P,()ab¡

b

a

y

O x

The -coordinate is always given

first. It indicates the movement

away from the origin in the

horizontal direction.

x

5

-4

-2

1

4

3

5 6

y

x

C(-4' 1)

A(3' 4)

B(5'-2)

O

For A , we say that:

is the of A

is the of A.

(3 4)

3

4

x

y

-coordinate

-coordinate

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Y:\HAESE\IGCSE01\IG01_12\256IGCSE01_12.CDR Thursday, 2 October 2008 12:36:28 PM PETER