Cambridge International AS and A Level Mathematics Pure Mathematics 1

Co-ordinate geometry

40

P1^

2

In general, when A is the point (x 1 , y 1 ) and B is the point (x 2 , y 2 ), the gradient is

m

yy

=xx

21

21

–

–.

When the same scale is used on both axes, m = tan θ (see figure 2.2). Figure 2.3

shows four lines. Looking at each one from left to right: line A goes uphill and

its gradient is positive; line B goes downhill and its gradient is negative. Line C is

horizontal and its gradient is 0; the vertical line D has an infinite gradient.

ACTIVITY 2.1 On each line in figure 2.3, take any two points and call them (x 1 , y 1 ) and (x 2 , y 2 ).

Substitute the values of x 1 , yl, x 2 and y 2 in the formula

m

yy

xx

=^21

21

–

–

and so find the gradient.

●?^ Does it matter which point you call (x 1 , y 1 ) and which (x 2 , y 2 )?

Parallel and perpendicular lines

If you know the gradients m 1 and m 2 of two lines, you can tell at once if they are

either parallel or perpendicular − see figure 2.4.

0  2 3   6  



2

3





y

x

A

B

C

D

Figure 2.3

m 1

m 1

m 2

m 2

Figure 2.4 parallel lines: m 1 =^ m 2 perpendicular lines: m 1 m 2 = −^1