Understanding Engineering Mathematics

B. Rewriting the lines in the form

y=mx+c

the gradient inm, the intercept on they-axis isc(x= 0 ,y=c)and

the intercept on thex-axis isx=−c/m,y=0.

(i)m=3,c=2, andx-intercept=− 2 / 3

(ii) y=^13 ( 1 − 2 x)

m=−^23 ,c=^13 andx-interceptx= 1 / 2

(iii)m=4,xandyintercepts both the origin (0, 0)

(iv)y=−x−1,m=−1, intercepts (0,−1), (−1, 0)

7.2.5 Parallel and perpendicular lines

➤

205 222➤

Consider two parallel lines,l 1 ,l 2 as shown in Figure 7.8.

y

l 1

l 2

0 x

N

P

M

B

A

Figure 7.8Parallel lines have the same gradient.

As we might expect, parallel lines have the same gradient (149

➤

). We can see this by

a nice application of the intercept theorem(153

➤

). The gradient ofl 1 is

AP

BP

and that of

l 2 is

MP

NP

. But, by the intercept theorem, sinceABandMNare parallel,MNdivides

APandBPin the same ratio, soAP/BP=MP/NP,i.e.l 1 ,l 2 have the same gradient.
Lines that are perpendicular clearly do not have the same gradient, but they are related.
If linesl 1 ,l 2 are perpendicular, and have gradientsm 1 ,m 2 respectively then

m 1 m 2 =− 1

We can see this from Figure 7.9.
The linesl 1 ,l 2 are perpendicular.
First, note that sinceα+θ=

π

2

=α+βthenβ=θ. If the gradient ofl 1 ism,then

m=

BD

AD

=tanθ