Coordinate Geometry

Make sure that you can find the equation of a straight line, using the form y – b = m(x – a) for a line
with gradient m through (a, b). Make sure you know how to find the coordinates of the midpoint of a
line segment joining two points and the distance between two points, and know the conditions for two
straight lines to be parallel or perpendicular.

Make sure you know the equation of a circle in the form (x – a)2 + (y – b)2 = r2 and can complete the
square to find the centre and radius of a circle. Check that you recall from GCSE: that a line from the
centre of the circle to the midpoint of a chord meets the chord at right angles and, conversely, the
perpendicular bisector of any chord of a circle passes through the midpoint of the circle; that the angle
in a semicircle is a right angle and, conversely, that, given any right-angled triangle, a circle can be
drawn passing through its vertices and with centre at the midpoint of the chord; and that the radius of
a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point.

Vectors
Note from the syllabus: “learners should distinguish carefully between vectors and scalars when
writing them by hand”. This means that you will lose marks if you don’t underline vectors in every case
where the notation is in bold on this sheet. Make absolutely sure you understand the distinction
between co-ordinates of points, position vectors and vectors as translations.

Make sure you know the notation xi + yj + zk as well as column vector notation; that a point P has
→ 2
co-ordinates e.g. (2, –5, 1) but position vector OP =  −5  , and the terms displacement vector,
 1
 
component vector, resultant vector, parallel vector, equal vector and unit vector. Make sure you know
how to find the magnitude or modulus of a vector in three dimensions and, in two dimensions, convert
between magnitude and direction (given as an anticlockwise angle of rotation from the positive x
direction between 0 and 360) and component form. Make sure you can use this to calculate the
distance between two points represented by their position vectors.
→ → →
1 Find the displacement BC if AB = −3i + 8 j + 5k and CA = −2i + 6 j − 7k .
2 Parallelogram ABCD has vertices A (–6, 3, –1), B (4, 6, –7) and C (12, –5, –3). Find |BD|.
→ →
3 If P is (1, –5, 7) and Q is (4, –1, 2), find a point X on line PQ with QX = 3PQ .
4 Find the point which divides AB in the ratio 3 : 1, where A is (5, –2, 3) and B is (–1, 1, 5).
5 AB is parallel to CD and |CD| = 21. If A is (1, 2, –5), B is (–2, 4, 1) and C is (4, –3, –5), find the
possible co-ordinates of D.
6 Show that in quadrilateral ABCD, with vertices A (–12, 4), B (–5, 13), C (6, 10) and D (2, –8), the
midpoint of AC lies on BD, and AB is perpendicular to BD. What is the area of this quadrilateral?
7 Given A (–7, 11), B (5, 3) and C (–2, –1), find, in the form ax + by + c = 0, where a, b and c are
integers, the equation of the line through C perpendicular to AB. Find the co-ordinates of the point
where this line meets AB, and hence find the area of triangle ABC.
8 Given A (–8, 12), B (–12, 10) and C (15, 1), find the equations of the perpendicular bisectors of
AB and BC. Hence find the equation of the circle which passes through A, B and C .

Find the equations of the tangents to the circle ( x − 3) + ( y + 1) = 45 which have gradient 2.
2 2
9

10 The points A (3, 4), B (–7, –2) and C (1, 6), lie on a circle. Find the gradients of AC and BC, and
hence find the equation of the circle.

11 Show that the tangent to the circle x 2 + y 2 + 8 x + 2y − 108 = 0 at the point (1, 9) passes through
A (7, 6). Find the co-ordinates of the points where the line through A perpendicular to the tangent
meet the circle.
 B
12
X

O Z
A
In the diagram, OAB is a triangle and OAZ is a straight line.
If it is given that
|OX| = 58 |OB|,
Y divides BA in the ratio 3 : 1,
|AZ| = 41 |OA|.
→ →
Let OA = a , OB = b ,
→
a Write XY in terms of a and b
b Prove that X, Y and Z are collinear.

13 A

P
Q

O B
→ →
In the diagram, let OA = a , OB = b .
Point Q divides AB in the ratio 1 : 2 and point P divides OA in the ratio 3 : 1
→
a Write PQ in terms of a and b.
→ →
Point R lies on line PQ and PR = t PQ .
b Write the position vector of R in terms of a, b and t.
c Point R also lies on line OB. Find the position vector of point R as a multiple of b.

14 The points A, C and X do not all lie in the same plane. O is the origin.
B is the fourth vertex of parallelogram OABC.
M is the midpoint of AX and P is the midpoint of BX.
→ → →
Let OA = a , OC = c and OX = x .
→ →
a Write CM and OP in terms of a, b and c.
b Show that CM and OP intersect and find the ratio in which the point of intersection divides OP.