Cambridge International AS and A Level Mathematics Pure Mathematics 1

The intersection of a line and a curve

P1^

2

(iii) Thelineandthecurvedonotmeet(see figure 2.36).

The co-ordinates of the point of intersection can be found by solving the two

equations simultaneously. If you obtain an equation with no real roots, the

conclusion is that there is no point of intersection.

The equation of the straight line is, of course, linear and that of the curve

non-linear. The examples which follow remind you how to solve such pairs of

equations.

EXAMPLE 2.14 Find the co-ordinates of the two points where the line y − 3 x = 2 intersects the
curve y = 2 x^2.

SOLUTION

First sketch the line and the curve.

x

y

y = x^2

y = x – 5

O 5

–5

Figure 2.36

O

y – 3x = 2

y = 2x^2

Figure 2.37