Cambridge International AS and A Level Mathematics Pure Mathematics 1

Co-ordinate geometry

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2

Different techniques to solve problems

The following examples illustrate the different techniques and show how these

can be used to solve a problem.

EXAMPLE 2.8 Find the equations of the lines (a) − (e) in figure 2.20.

SOLUTION

Line (a) passes through (0, 2) and has gradient 1

⇒ equation of (a) is y = x + 2.

Line (b) is parallel to the x axis and passes through (0, 4)

⇒ equation of (b) is y = 4.

Line (c) is parallel to the y axis and passes through (−3, 0)

⇒ equation of (c) is x = −3.

Line (d) passes through (0, 0) and has gradient − 2

⇒ equation of (d) is y = − 2 x.

Line (e) passes through (0, −1) and has gradient –^15

⇒ equation of (e) is yx=––^151.

This can be rearranged to give x + 5 y + 5 = 0.

EXAMPLE 2.9 Two sides of a parallelogram are the lines 2y = x + 12 and y = 4 x − 10. Sketch

these lines on the same diagram.

The origin is a vertex of the parallelogram. Complete the sketch of the

parallelogram and find the equations of the other two sides.

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2
1
3
4
5
y
x
–2
–3
–2–1
(a)
(b)
(e)
(d)
(c)
–3 –1
Figure 2.20