O LEVEL (P1)
COORDINATE
GEOMETERY
QUESTION'S
1
1
y
R
4
3
2
1
P
–4
Q
–3
–2
–1
0
1
2
3
4
5
6
7
8
9 x
–1
PQRS is a parallelogram.
P is (–4, 0), Q is (1, 0) and R is (9, 4).
(a) Find the coordinates of S.
(b) Find the coordinates of the midpoint of PR.
(c) Find the equation of the line RS.
(d) Find the equation of the line QR.
(e) Calculate the area of the parallelogram PQRS.
Answer (a)
(............ , ............)
[1]
(b)
(............ , ............)
[1]
(c) ............................................ [1]
(d) ........................................... [2]
(e) ....................................unit2 [1]
2
2
The diagram shows the points A (1, 2), B (4, 6) and D (–5, 2).
y
B(4,6)
D(–5,2)
A(1,2)
x
0
(a) Find the coordinates of the midpoint of AB.
(b) Calculate the length of AB.
(c) Calculate the gradient of the line AB.
(d) Find the equation of the line AB.
(e) The triangle ABC has line of symmetry x = 4.
Find the coordinates of C.
(f)
Find the value of cosine DÂB.
Answer (a)
(.................. , ..................)
[1]
(b) ................................................... [1]
(c) ....................................................[1]
(d) ....................................................[2]
(e)
(.................. , ..................)
[1]
(f) cos DÂB = .................................[1]
4024/01/M/J/06
3
3
y
10
8
6
A
4
2
–4
–2
0
–2
2
4
C
6
8
x
10
–4
B
–6
The triangle with vertices A(4, 4), B(–2, –6) and C(4, –1) is shown in the diagram.
Find
(a) (i) the area of ∆ ABC,
(ii) the coordinates of the point P such that ABCP is a parallelogram,
(iii) the area of the parallelogram ABCP,
^
(iv) tan BAC.
(b) It is given that the length of BC = k units.
^
Write down cos BCA, giving your answer in terms of k.
Answer (a) (i) .................................unit2 [1]
(ii)
(............ , ............)
[1]
(iii) .................................unit2 [1]
^
(iv) tan BAC = ........................[1]
^
(b) cos BCA = ............................[1]
___________________________________________________________________________
4
4 A straight line passes through the points P (1, 2) and Q (5, –14).
Find
(a) the coordinates of the midpoint of PQ,
(b) the gradient of PQ,
(c) the equation of PQ.
Answer (a)
(............. , .............)
[1]
(b) ............................................[1]
(c) ............................................[2]
5
5 (a) Find the coordinates of the point where the line 2y = 3x + 15 crosses the y-axis.
Answer (a)
(............ , ............)
[1]
(b) The coordinates of the points P and Q are (–1 , 10) and (3 , 4) respectively.
Find
(i) the gradient of PQ,
Answer (b)(i) ...................................[1]
(ii) the midpoint of PQ.
Answer (b)(ii)
(............ , ............) [1]
6
6 A straight line passes through the points P (–8, 10) and Q (4, 1).
Find
(a) the coordinates of the midpoint of PQ,
Answer (a) (............... , ................) [1]
(b) the equation of PQ.
Answer (b) ..................................... [2]
7
7 M is the midpoint of the line joining P and Q.
(a) R lies on PQ produced, such that PR = 3PQ .
Find PM : PR .
Answer (a) .................. : ..................[1]
(b) P is ( 1, –2 ) and Q is ( 5, 6 ) .
(i) Find the coordinates of M.
Answer (b)(i) (............. , .............) [1]
(ii) The line 4x + ky + 10 = 0 passes through Q ( 5, 6 ) .
Find the value of k.
Answer (b)(ii) k = ............................ [2]
8
8 P is the point (–2, 1) and Q is the point (3, 7).
(a) M is the midpoint of PQ.
Find the coordinates of M.
Answer
(............, ............)
[1]
Answer
........................................ [1]
(b) Find the gradient of the line PQ.
(c) The line with equation 2y + 3x + k = 0 passes through the point P.
(i) Find k.
Answer
k = .................................. [2]
Answer
........................................ [1]
(ii) Find the gradient of this line.
9
9 Here are the equations of four straight lines.
Line 1:
Line 2:
Line 3:
Line 4:
y = 2x + 4
y = 2 – x
y = 2x – 1
2y – 8 = 3x
(a) Which two lines are parallel?
Answer Line ........... and Line ........... [1]
(b) Which two lines intersect the y-axis at the same point?
Answer Line ........... and Line ........... [1]
(c) Which line passes through the points (1, 1) and (–3, 5)?
Answer Line ............................. [1]
(d) Find the midpoint of the line segment joining (1, 1) and (–3, 5).
Answer (................ , ................) [1]
10
10 A is the point (0, 4) and B is the point (–6, 1).
(a) M is the midpoint of the line AB.
Find the coordinates of M.
Answer
(............, ............)
[1]
Answer
..................................... [2]
(b) Find the equation of the line AB.
11 A line has equation 3y = 2 – x �
(a) Find the gradient of the line�
Answer
�������������������������������������������� [1]
(b) The line passes through the point (5, k)�
Find the value of k�
Answer k = �������������������������������������� [1]
11
12
P is (−1, 3) and Q is (5, −1).
(a) Find the coordinates of the midpoint of PQ.
Answer (............... , ..............) [1]
(b) Find the gradient of the line PQ.
Answer
............................................... [1]
(c) Given that the length of PQ = 2 n units, where n is an integer, find the value of n.
Answer n = ........................................ [2]
12
13 (a) The line 2y = 6 - 3x meets the y-axis at A and the x-axis at B.
Write down
(i) the coordinates of A and B,
Answer
A = ( ................ , ................)
B = ( ................ , ................ ) [2]
(ii) the gradient of the line.
Answer
.......................................................... [1]
(b) Another straight line cuts the x-axis at P ^- 4, 0h and passes through Q ^2, 18h.
Find the coordinates of the midpoint of PQ.
Answer
( .......................... , ......................... ) [1]
13
14
y
8
6
4
2
–2
0
2
4
6
8
10 x
–2
–4
(a) On the grid above, draw the graph of
x+y = 6.
[1]
(b) On the grid above, draw the graph of
2y + x = 4 .
[1]
(c) On the grid above, shade and label the region R, defined by the following inequalities.
x+y G 6
2y + x H 4
yH2
x H-1
[2]
14
15 P is ^-4, 4h and Q is ^3, -2h.
M is the midpoint of PQ.
(a) Find the coordinates of M.
Answer ( ................... , ...................) [1]
(b) Find the gradient of the line PQ.
Answer
............................................. [1]
(c) Q is the midpoint of the line PQR.
(i) Find the coordinates of R.
Answer ( .................... , ..................) [2]
(ii) Write down the value of
PM
.
MR
Answer
............................................. [1]
15
16 P is the point (1, –3) and Q is the point (7, 2).
(a) Find the coordinates of the midpoint of PQ.
Answer ( .................... , .................... ) [1]
(b) Find the gradient of the line PQ.
Answer ................................................. [1]
(c) The line, L, with equation 2x – 5y = k, passes through the point Q.
(i) Find the value of k.
Answer k = ................................................. [1]
(ii) The line x + Ay = 3 is parallel to L.
Find the value of A.
Answer A = ................................................. [1]
17 The coordinates of the midpoint of the line AB are16(1, 2).
The length of the line AB is 10 units.
(a) If the gradient of AB is 0, find the coordinates of A and B.
Answer A = (............ , ............)
B = (............ , ............)
(b) If the gradient of AB is
[1]
3
, find the coordinates of A and B.
4
Answer A = (............ , ............)
B = (............ , ............)
[2]
18 A is the point (0, 3), B is the point (1, 5) and C is the point (p, –1).
(a) Find the equation of the line AB.
Answer ........................................... [2]
(b) The gradient of the line BC is -
3
.
4
Find the value of p.
Answer p = ..................................... [2]