Cambridge International AS and A Level Mathematics Pure Mathematics 1

Co-ordinate geometry

P1^

2

The intersection of two lines

The intersection of any two curves (or lines) can be found by solving their

equations simultaneously. In the case of two distinct lines, there are two

possibilities:

(i) they are parallel

(ii) they intersect at a single point.

EXAMPLE 2.11 Sketch the lines x + 2 y = 1 and 2x + 3 y= 4 on the same axes, and find the

co-ordinates of the point where they intersect.

SOLUTION

The line x + 2 y = 1 passes through () 0 ,^12 and (1, 0).

The line 2x + 3 y = 4 passes through () 0 ,^43 and (2, 0).

^1 : x^ +^2 y^ =^1 ^1 :^ × 2: 2x^ +^4 y^ =^2

^2 : 2x^ +^3 y^ =^4 ^2 : 2x^ +^3 y^ =^4

Subtract: y = −2.

Substituting y = −2 in ^1 : x − 4 = 1

⇒ x = 5.

The co-ordinates of the point of intersection are (5, −2).

O 1 2

x + 2y = 1

2 x + 3y = 4

x

y

-^12
-^43

Figure 2.23