HORIZON EDUCATION SINGAPORE
Additional Mathematics
Practice Questions: Coordinate Geometry 1
Set 1
1
In the figure, 𝐴𝐵𝐶𝐷 is a rhombus with coordinates 𝐴(2, 9) and 𝐶(8, 1).
The diagonals 𝐴𝐶 and 𝐵𝐷 cut at 𝐸.
(i)
Calculate the co-ordinates of 𝐸.
[1]
(ii)
Find the equation of 𝐵𝐷.
[2]
It is given that the equation of 𝐴𝐷 is 𝑥 + 7𝑦 − 65 = 0.
(iii) Find the equation of 𝐵𝐶.
(iv)
2
3
4
Calculate the length of 𝐴𝐵.
[2]
[3]
The line 3𝑥 − 𝑦 = 7 intersects the curve 𝑥 2 − 𝑥𝑦 + 𝑦 2 = 7 at 𝐴 and 𝐵.
Find
(a) the coordinates of the points 𝐴 and 𝐵,
[3]
(b) the equation of the perpendicular bisector of 𝐴𝐵.
[3]
1
The line 𝑦 = 2𝑥 intersects the curve 𝑦 = 𝑥 + 𝑥 at points 𝐴 and 𝐵. Find the
equation of the perpendicular bisector of the line 𝐴𝐵.
[6]
Two points 𝐴 and 𝐵 have coordinates (3√2, 2√5) and (−2√5, √2) respectively.
Without the use of a calculator, calculate the gradient of the line 𝐴𝐵, leaving your
answer in the form 𝑎 + 𝑏√10, where 𝑎 and 𝑏 are integers.
[4]
Page 1 of 12
Division of Mathematics, Horizon Education Singapore
5
Solution to this question by accurate drawing will not be accepted.
y
B (6, 13)
C
A
3 y  4 x  10
x
D (1,
)
The diagram shows a trapezium 𝐴𝐵𝐶𝐷 in which 𝐴𝐵 is parallel to 𝐷𝐶. The point 𝐴
lies on the 𝑦-axis. Points 𝐵 and 𝐷 are (6, 13) and (1, −2) respectively. 𝐴𝐵̂ 𝐶 =
𝐵𝐶̂ 𝐷 = 90°. Given that the equation of 𝐷𝐶 is 3𝑦 = 4𝑥 − 10, find
(a) the coordinates of 𝐴,
[3]
6
(b) the coordinates of 𝐶,
[4]
(c) the area of the trapezium 𝐴𝐵𝐶𝐷.
[3]
Solutions to this question by accurate drawing will not be accepted.
The coordinates of 𝑃 and 𝑄 are (−1, 10) and (11, 6) respectively.
(i)
Find the equation of the perpendicular bisector of 𝑃𝑄.
[3]
(ii)
(iii)
Given that there is a pair of coordinates of point 𝐴 which meets the
perpendicular bisector of 𝑃𝑄 at the 𝑦-axis, find coordinates of 𝐴.
[3]
If 𝑃𝐴𝑄𝐵 is a parallelogram, find the coordinates of point 𝐵.
[3]
Page 2 of 12
Division of Mathematics, Horizon Education Singapore
7
Solutions to this question by accurate drawing will not be accepted.
y
A(7, 10)
B
. M(8, 6)
D
C
x
The diagram shows a rhombus 𝐴𝐵𝐶𝐷 in which 𝐴 is (7, 10) and 𝐷 is on the 𝑦-axis.
The point 𝑀(8, 6) is the midpoint of 𝐴𝐶. Find
(a) the coordinates of 𝐶,
[2]
(b) the coordinates of 𝐷,
[3]
(c) the coordinates of 𝑋 given that 𝑋 is on 𝑀𝐷 such that area of triangle 𝐴𝑋𝐶 =
1
area of triangle 𝐴𝐷𝐶.
[3]
4
8
Solutions to this question by accurate drawing will not be accepted.
y
C (2, 1)
B
x
0
D
A (−1, −4)
In the quadrilateral 𝐴𝐵𝐶𝐷, the points 𝐴 and 𝐶 are (−1, −4) and (2, 1)
respectively. The line 𝐵𝐶 is parallel to 𝑥 − 4𝑦 − 1 = 0 and perpendicular to 𝐴𝐵.
The foot of the perpendicular from 𝐴 to 𝐶𝐷 bisects 𝐶𝐷 and the rest on the 𝑥-axis
with the 𝑥-coordinate 3.
Find
(i)
the coordinates of 𝐵 and of 𝐷,
(ii)
the radius of the circle which passes through 𝐴, 𝐵 and 𝐶.
Page 3 of 12
Division of Mathematics, Horizon Education Singapore
[8]
9
Solution to this question by accurate drawing will not be accepted.
𝑃𝑄𝑅𝑆 is a trapezium in which 𝑃𝑄 is parallel to 𝑆𝑅 and 𝑃𝑆 is perpendicular to
both 𝑃𝑄 and 𝑆𝑅. The coordinate of 𝑃, 𝑄 and 𝑅 are (0, 11), (3, 2) and (13, 12)
respectively.
y
S
R (13, 12)
P(0, 11)
Q(3, 2)
10
x
Find
(a) the equation of 𝑆𝑅,
[2]
(b) the coordinates of the point 𝑆,
[2]
𝑇 is a point on 𝑅𝑆 produced such that 𝑃𝑄𝑅𝑇 is a parallelogram. Find
(c) the coordinates of 𝑇,
[2]
(d) the ratio 𝑅𝑆: 𝑆𝑇,
[2]
(e) the shortest distance of 𝑅 from 𝑃𝑄.
[2]
𝐴𝐵𝐶𝐷 is a rectangle, where 𝐴 is (−3, 0) and 𝐶 is (1, 7).
Given that the equation of 𝐴𝐵 is 3𝑦 = 2𝑥 + 6, find
(i)
the equation of 𝐵𝐶,
[2]
(ii)
the coordinates of 𝐵,
[2]
(iii)
the coordinates of 𝐷,
[2]
(iv)
the area of 𝐴𝐵𝐶𝐷.
[2]
Page 4 of 12
Division of Mathematics, Horizon Education Singapore
11
The diagram, which is not drawn to scale, shows a right-angled triangle 𝑃𝑄𝑅 in
which ∠𝑄𝑃𝑅 = 90° and the coordinates of 𝑃 and 𝑄 are (3, 5) and (−1, −3)
1
respectively. Given that the gradient of 𝑄𝑅 is 2 and the the perpendicular from 𝑃
to 𝑄𝑅 to meets 𝑄𝑅 at 𝑆, find
P (3, 5)
y
R
S
x
O
Q(1, 3)
(a) the equation of 𝑄𝑅 and of 𝑃𝑅,
[3]
(b) the coordinates of 𝑅 and of 𝑆,
[5]
(c) the ratio, 𝑄𝑆: 𝑆𝑅,
[1]
area of Δ𝑃𝑆𝑅
(d) the numerical value of area of Δ𝑃𝑄𝑅.
Page 5 of 12
Division of Mathematics, Horizon Education Singapore
[1]
12
In the quadrilateral 𝐴𝐵𝐶𝐷, the points 𝐴, 𝐵 and 𝐷 are (3, 3), (0, −1) and (6, 2)
respectively. The line 𝐵𝐷 bisects the line 𝐴𝐶 at right angles at the point 𝑀. Find
the coordinates of 𝑀 and of 𝐶.
[8]
𝑦
𝐴 (3, 3)
𝐷 (6, 2)
𝑀
𝑥
𝐵
(0, −1)
13
𝐶
The straight line 𝑦 + 2𝑥 = 5 intersects the curve 𝑥 2 + 𝑦 2 + 𝑥 + 12𝑦 = 29 at the
points 𝐴 and 𝐵. Given that 𝐴 lies below the 𝑥-axis, and that 𝑃 lies on 𝐴𝐵 such that
1
the area of Δ𝐴𝑂𝑃 is 4 of the area of Δ𝐴𝑂𝐵, where 𝑂 is the origin, find the
coordinates of 𝑃.
[6]
Page 6 of 12
Division of Mathematics, Horizon Education Singapore
14
Solutions to this question by accurate drawing will not be accepted.
The diagram shows a kite 𝑂𝐴𝐵𝐶 whose diagonals meet at 𝑀. The coordinates of
𝐴, 𝐵 and 𝐶 are (11, 𝑎), (5, 3) and (𝑐, 𝑐 + 3) respectively, where 𝑎 and 𝑐 are
y
constants.
C(c, c + 3)
B(5, 3)
M
x
O
A(11, a)
15
16
Find
(a) the coordinates of 𝑀,
[1]
(b) the equation of 𝐴𝐶,
[3]
(c) the value of 𝑎,
[1]
(d) the coordinates of 𝐶,
[2]
(e) the area of the kite 𝑂𝐴𝐵𝐶.
[3]
𝑃, 𝑄, 𝑅 and 𝑆 are the points (0, 9), (−3, −3), (1, −2) and (3, 7) respectively.
(i)
Find the area of the quadrilateral 𝑃𝑄𝑅𝑆.
[2]
(ii)
Find the equation of the line, 𝑙1, that is perpendicular to 𝑃𝑆 and passes
through the point 𝑅.
[3]
(iii)
If 𝑃𝑄 is extended to meet the line 𝑙1 at 𝑇, find the ratio of 𝑃𝑄: 𝑄𝑇.
[4]
A point 𝑀 lies on the line 2𝑦 + 𝑥 = 10 and is at a distance of 5 units from the
origin (0, 0). Find the possible 𝑥-coordinates of 𝑀.
[3]
Page 7 of 12
Division of Mathematics, Horizon Education Singapore
17
18
𝐴𝐵𝐶𝐷 is a rectangle such that 𝐴, 𝐵 and 𝐶 are the points (0, 1), (𝑡, 2𝑡 + 1) and
(4, 4) respectively.
(i)
Show that the value of 𝑡 is 2.
[2]
(ii)
Find the equation of 𝐶𝐷.
[2]
(iii)
Find the coordinates of the point of intersection of the 2 diagonals.
[2]
(iv)
Find the equation of the perpendicular bisector of 𝐴𝐵.
[3]
Solutions to this question by accurate drawing will not be accepted.
The line 4𝑥 − 3𝑦 = 1 intersects the curve 𝑥𝑦 = 28𝑦 − 27𝑥 at the points 𝑃 and 𝑄.
(a) Find the coordinates of 𝑃 and of 𝑄.
[4]
(b) Find the equation of the perpendicular bisector of 𝑃𝑄.
[3]
It is given that the perpendicular bisector of 𝑃𝑄 intersects the 𝑦-axis at the point
𝑅.
(c) Find the distance of 𝑅 from 𝑃𝑄.
[3]
19
In the given diagram, point 𝐴 is located on the 𝑦-axis and the equation of the
straight line 𝐴𝐵 is 2𝑦 = 𝑥 + 8.
y
B (p, q)
A
O
x
(a) Write the coordinates of the midpoint of 𝐴𝐵 in terms of 𝑝 and 𝑞.
[2]
(b) Show that the equation of the perpendicular bisector of 𝐴𝐵 is 2𝑦 + 4𝑥 = 2𝑝 +
𝑞 + 4.
[2]
(c) If the perpendicular bisector of 𝐴𝐵 intersects the 𝑦-axis at 𝑦 = 14, find the
value of 𝑝 and of 𝑞.
[3]
20
The line 5𝑥 + 𝑦 = 9 intersects the curve 𝑦 2 + 3𝑥𝑦 = −5 at the points 𝑃 and 𝑄.
Find the midpoint of 𝑃𝑄.
[6]
Page 8 of 12
Division of Mathematics, Horizon Education Singapore
21
Solution to the question by accurate drawing will not be accepted.
y
A (2, 9)
B
y = 2x
x
O
The diagram shows a right-angled triangle 𝐴𝐵𝑂 in which 𝑂 is the origin, 𝐴 is the
point (2, 9) and 𝐴𝐵̂ 𝑂 = 90°. The equation of the line 𝑂𝐵 is 𝑦 = 2𝑥.
Find
(i)
the equation of the line 𝐴𝐵,
[2]
(ii)
the coordinates of 𝐵.
[2]
𝐶 is a point on the perpendicular bisector of 𝑂𝐴 and is such that 𝐵𝐶 is parallel to
the 𝑦-axis.
(iii) Find the coordinates of 𝐶.
[3]
area of Δ𝑂𝐴𝐵
1
𝐷 lies on 𝐴𝐵 produced such that area of Δ𝑂𝐴𝐷 = 3.
(iv)
Find the coordinates of 𝐷.
Page 9 of 12
Division of Mathematics, Horizon Education Singapore
[3]
22
Solutions to this question by accurate drawing will not be accepted.
The diagram shows a trapezium 𝐴𝐵𝐶𝐷 in which the coordinates of 𝐴 and 𝐶 are
(2, −1) and (3, 4) respectively Given that 𝐸 is a point on the 𝑦-axis, such that
𝐴𝐵𝐶𝐸 is a square.
y
C (3, 4)
D
E
B
x
O
A(2, 1)
(i)
Find the coordinates of 𝐸 and 𝐵.
[4]
(ii)
Find the equation of 𝐴𝐵.
[2]
(iii)
Given that the area of square 𝐴𝐵𝐶𝐸 is 4 times the area of triangle 𝐶𝐷𝐸,
find the coordinates of 𝐷.
[3]
Page 10 of 12
Division of Mathematics, Horizon Education Singapore
23
Solutions to this question by accurate drawing will not be accepted.
y
A (6, 9)
C (8,4)
B (6, 1)
0
7
E (p, -2
x
)
13
D
The diagram shows an isosceles triangle 𝐴𝐵𝐶 in which 𝐴 is the point (6, 9). 𝐵 is
the point (6, 1). It is given that the area of the triangle 𝐴𝐵𝐶 is 8 units2.
(i)
Find the coordinates of 𝐶.
[2]
The line 𝐶𝐵 is extended to the point 𝐷 such that the area of triangle 𝐴𝐷𝐶 is thrice
the area of triangle 𝐴𝐵𝐶.
(ii)
Find the coordinates of 𝐷.
[3]
7
A line is drawn from 𝐴, parallel to 𝐶𝐷, to the point 𝐸 (𝑝, −2 13).
(iii)
Find the value of 𝑝.
[2]
(iv)
Determine whether ∠𝐴𝐸𝐷 = 90°.
[3]
Page 11 of 12
Division of Mathematics, Horizon Education Singapore
24
Solutions to this question by accurate drawing will not be accepted.
The diagram shows a quadrilateral 𝐴𝐵𝐶𝐷 in which 𝐴 is (3, 0), 𝐶 is (3𝑎, 5𝑎 + 3)
and 𝐷 is (−2, 5). The equation of 𝐴𝐵 is 5𝑦 = 3𝑥 − 9 and angle 𝐴𝐷𝐶 = 90°.
y
C (3a, 5a + 3)
D (– 2, 5)
B
x
A (3, 0)
(i)
Find the value of 𝑎.
If 𝐹 is the foot of the perpendicular bisector of 𝐶𝐷 from 𝐵, find
(ii)
the coordinates of 𝐹,
(iii)
the coordinates of 𝐵, and
(iv)
the area of the quadrilateral 𝐴𝐵𝐶𝐷.
[11]
25
The line 2𝑦 + 𝑥 = 5 intersects the cruve 𝑦 2 + 𝑥𝑦 = 6 at the points 𝐴 and 𝐵. Find
the equation of the perpendicular bisector of 𝐴𝐵.
[6]
26
The points 𝐴 and 𝐵 have coordinates (−8, −10) and (−2, −2). Find the equation
of the perpendicular bisector of 𝐴𝐵.
[4]
END
Page 12 of 12
Division of Mathematics, Horizon Education Singapore