CAIE A-Level Mathematics 1.3 Coordinate geometry Question Bank

CAIE A-Level Mathematics 1.3 Coordinate geometry Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise coordinate geometry with straight lines, circles, tangents and intersections by forming equations from points and graphs.

Exam points

  • find line equations from gradients, points, midpoints or perpendicular relationships
  • use circle centres and radii to write equations or find tangent equations
  • eliminate variables to test intersections with a quadratic discriminant or solve coordinates

Covered syllabus points

1.3.1—Straight lines1.3.2—Line equation forms1.3.3—Circle equation1.3.4—Lines and circles1.3.5—Graph intersections

Question 1

[Maximum number: 5]

A circle has centre (2,6) and radius 10.

Question 1(a)

(a)

State the equation of the circle.

[ 2 ]

Question 1(b)

(b)

The circle passes through the point (8, k).

Find the two possible values of k.

[ 3 ]

Question 6

[Maximum number: 9]

The equation of a curve is 2x2−kxy+2=02 x^{2}-k x y+2=0 and the equation of a line is y=p x+3, where k and p are constants.

Question 6(a)

(a)

Given that k=2 and p=11, find the coordinates of the points of intersection of the curve and the line.

[ 4 ]

Question 6(b)

(b)

（b）Given instead that p=4 ，find the set of values of k for which the curve and the line do not intersect．

[ 5 ]

Question 11(a)

[Maximum number: 2]
Figure for Question 11(a) — CAIE A-Level Mathematics

The diagram shows the curve with equation x=y2+1x=y^{2}+1. The points A(5,2) and B(2,-1) lie on the curve.

Find an equation of the line A B.

Question 10

[Maximum number: 7]

A circle has equation x2+y2+4y−21=0x^{2}+y^{2}+4 y-21=0 and a straight line has equation 2 x+y-8=0. The line intersects the circle at two points.

Question 10(a)

(a)

Find the coordinates of these two points of intersection.

[ 4 ]

Question 10(b)

(b)

The circle has centre C and the two points of intersection are denoted by A and B.

Find the area of the triangle ABC.

[ 3 ]

Question 11

[Maximum number: 7]

The coordinates of points A, B and C are (6, 4), ( p, 7 ) and (14, 18) respectively, where p is a constant. The line A B is perpendicular to the line B C.

Question 11(a)

(a)

Given that p<10, find the value of p.

[ 4 ]

Question 11(c)

(b)

Find the equation of the tangent to the circle at C, giving the answer in the form d x+e y+f=0, where d, e and f are integers.

[ 3 ]