Cambridge International AS and A Level Mathematics Pure Mathematics 1

Co-ordinate geometry

P1^

2

(i) Given the gradient, m, and the co-ordinates (x 1 , y 1 ) of one

point on the line

Take a general point (x, y) on the line, as shown in figure 2.15.

The gradient, m, of the line joining (x 1 , y 1 ) to (x, y) is given by

m

yy

=xx

–

–

1

1

⇒ y− y 1 = m (x− x 1 ).

This is a very useful form of the equation of a straight line. Two positions of the

point (x 1 , y 1 ) lead to particularly important forms of the equation.

(a) When the given point (x 1 , y 1 ) is the point (0, c), where the line crosses the

y axis, the equation takes the familiar form

y = mx + c

as shown in figure 2.16.

(b) When the given point (x 1 , y 1 ) is the origin, the equation takes the form

y = mx

as shown in figure 2.17.

y

x

x y

x y

O

Figure 2.15

y

x

y = mx

O

y

x

y = mx + c

(0, c)

O

Figure 2.16 Figure 2.17