Cambridge International AS and A Level Mathematics Pure Mathematics 1

Co-ordinate geometry

P1^

2

7 The equation of a curve is xy = 10 and the equation of a line l is 2x + y = q,

where q is a number.

(i) In the case where q = 9, find the co-ordinates of the points of intersection

of l and the curve.

(ii) Find the set of values of q for which l does not intersect the curve.

8 The curve y^2 = 12 x intersects the line 3y = 4 x + 6 at two points. Find the

distance between the two points.

[Cambridge AS & A Level Mathematics 9709, Paper 1 Q5 June 2006]

9 Determine the set of values of the constant k for which the line y = 4x + k

does not intersect the curve y = x^2.

[Cambridge AS & A Level Mathematics 9709, Paper 1 Q1 November 2007]

10 Find the set of values of k for which the line y = kx − 4 intersects the curve

y = x^2 − 2 x at two distinct points.

[Cambridge AS & A Level Mathematics 9709, Paper 1 Q2 June 2009]

KEY POINTS

1 The gradient of the straight line joining the points (x 1 , y 1 ) and (x 2 , y 2 ) is

given by

gradient =

yy

xx

21

21

–

 –.

when the same scale is used on both axes, m = tan θ.

2 Two lines are parallel when their gradients are equal.

3 Two lines are perpendicular when the product of their gradients is −1.

4 When the points A and B have co-ordinates (x 1 , y 1 ) and (x 2 , y 2 ) respectively,

then

the distance AB is ()xx 21 −+^2 ()yy 21 −^2

the mid-point of the line AB is xx 12 yy 12

22

 ++







,.

5 The equation of a straight line may take any of the following forms:

● line parallel to the y axis: x = a

● line parallel to the x axis: y = b

● line through the origin with gradient m: y = mx

● line through (0, c) with gradient m: y = mx + c

● line through (x 1 , y 1 ) with gradient m: y − y 1 = m(x − x 1 )

● line through (x 1 , y 1 ) and (x 2 , y 2 ):

yy

yy

xx

xx

yy

xx

yy

xx

–

–

–

–

–

–

–

–.

1

21

1

21

1

1

21

21

= or =