Cambridge International AS and A Level Mathematics Pure Mathematics 1

Co-ordinate geometry

P1^

2

The intersection of a line and a curve

When a line and a curve are in the same plane, there are three possible situations.

(i) Allpointsofintersectionaredistinct (see figure 2.34).

(ii) Thelineisatangenttothecurveatone(ormore)point(s) (see figure 2.35).

In this case, each point of contact corresponds to two (or more) co-incident

points of intersection. It is possible that the tangent will also intersect the curve

somewhere else.

x x

y y

1

1

y = x^2

y = x + 1

x + 4y = 4

(x – 4)^2 + (y – 3)^2 = 2^2

O O

Figure 2.34

x

x

y

y

y = 1

(–2, 8)

y = 2x + 12

y = x^3 + x^2 – 6x

(x – 4)^2 + (y – 4)^2 = 3^2

O

O

• 3 2

12

Figure 2.35