CHAPTER 8

l.Za COORIIINATE GEOMETRY

":u are already familiar with using co-ordinates in nvo dimensions. A frame of reference is set up by using wonr.;rually perpendicular lines (the coordinate axes) intersecting at the point O (ttre origin). Points in the coordinate

tilure are represented by their perpendicular distances from the coordinate axes.

-me coordinates are often known as Carteslan co-ordinates (after the French philosopher and mathematiciantcae Descartes) to distinguish them from other representations such as polar coordinates. Traditionally the axes

re drawn so that the y-axis is at 90o to ftre r-axis measured anti-cloclsilise.

..r..r.(r.tr.r r.r rr. r.. r..g n.r t.ir.rr.tt.(t.rtr """""""fP{a,b)

!:t

ibi

:,

!a

The co-ordinates of the point P are written as

an ordered pair in a horizontal bracket with the

.r-coordinate (abscissa) first, followed by the

y-coordinate (ordinate).

(distances in the direction of the &rows on the

axes are positive, and negative in the opposite

direction.)

1e system can be extended to three dimensions by adding a z-axis which is perpendicular to bottr the

:-rxis and the y-axis. This is traditionally drawn to form a right-hand screw - if the x-axis points south and the

-rxis points east then thez-axis points vertically upwards. Two common ways of representing this in a 2-dimen-

'rr:nal diagram are

kpointP (a, b, c) isthen a distance a from O intherdirection,6 in they direction andc inthez direction.

99

Chapter 8: Coordinate Geometry

Twel)ftrr,xsronal, Coonon,qrr GnoMrrnyAny straight line in the Cartesian plane can be represented by the equation y: mx + c where n represents the

gradient and c the intercept on the y-axis.

Remembeq that if you are using the equation to find the intercept and gradient, you must first make

ythesubject. Iftheequationofaline is 2y+3x:S,thegradientisnot3but -9sincewith y as subject the equation of the line is y: -tx + 4.

Notice, also, the'ways in which we usually refer to three types of points:

(i) Fixed points whose coordinates ar:e known: (5, -7) etc.

(ii) Fixed points whose coordinates me not known numesricall y: (a, b) or (x1, !r), (xz,yj etc.

(iii) General points, which are not fixed: (x, y).

Given the gradient and intercept of a line, you can wrjte down its equation; but you are often working with other

information than gradients and/or intercepts. The next section reminds you, with proofs, how to find the gradi-

ent, the mid-point, the equation and the length of a line given any two points A (xr,.yr) and B (x2, y2) on it.

The gradient of AB

If P (x, y) is any point on the line AB, then the gradient of AP is also m, so equating expressions for m

The equation of AB can be written

alternatively, since ,n * f,,:,!1,f, *Jft

The distance AB can be found

using S'thagoras' theorenr:

L-l:ii:dI-:

I n*, : i., *.,1, * fr* *rnl, I

A(xr,yr)

4z:,,I"\xz* xt

Iz * -vtxz* xt

100

Pure Mathematics for CAPE

If M (1 n) is the mid-point of AB

r:*t+*kz-.rr): ry and

.(ry,ry)

When doing coordinate geometry, do not use a calculator. Gradients are often fractions, but they

should not be tumed into decimals - either leave them as fractions, or nnrltiply both sides of an equa-

tion by the denominator so that you work with integers.

A sketch often helps to visualise aproblem, and can also indicate when an answer is obviously wrong.