20181122 F.6 MC Practice (Coordinate Geometry)
Level 1: single concept, simple calculation, easier than HKDSE types
Level 2: one or two concept, some calculations, similar to HKDSE easy types
Level 3: involving high level, logical and abstract thinking skills, or with
complicated calculations, similar to HKDSE difficult types
** Please answer any 30 questions **
Level 1
1.
Find the distance between A(−3 , 4) and B(2 , −6), correct to 3 significant figures.
A. 2.24 units
B. 5.04 units
C. 6.32 units
D. 11.2 units
2.
A(−4 , 1) and B(3 , 5) are points on a straight line L. Find the slope of L.
A.
3.
4
7
B.
−
C.
7
4
D.
−
7
4
L1, L2 and L3 are three straight lines. If L1 // L2, L2 ⊥ L3 and the slope of L1 is
find the slope of L3.
4
A. −
B.
3
4.
4
7
−
3
4
C.
−1
D.
3
,
4
3
4
The coordinates of A and B are (8 , 3) and (−1 , 2) respectively. If a straight line L
is parallel to AB, find the inclination of L, correct to the nearest 0.1°.
A. 6.3°
5.
B.
14.0°
C.
20.6°
D.
83.7°
A(5 , 1) and B(3 , 9) are two given points. If M is the mid-point of the line
segment AB, find the coordinates of M.
A. (8 , 10)
B.
(4 , 5)
C.
(2 , −8)
D.
(1 , −4)
6.
y
E(−3 , 1)
H(2 , 0)
O
F(−2 , −2)
x
G
In the figure, EFGH is a parallelogram. Find the coordinates of G.
A. (3 , −1)
7.
B.
(3 , −3)
C.
(4 , −3)
D.
(4 , −4)
If M(–1 , –4), N(3 , 0) and P(1 , y) are three points on the same straight line, then
y=
A. 4.
B.
2.
C.
1
–4.
D.
–2.
20181122 F.6 MC Practice (Coordinate Geometry)
8.
L3
In the figure, the slopes of straight lines L1, L2 and L3 are m1, m2 and m3
respectively. Which of the following must be correct?
I.
A.
C.
9.
m1 < 0
II.
I only
I, II and III only
m2 < 0
III. m3 > 0
IV. m2 > m1
B.
II and III only
D.
I, II, III and IV
Three points A(3 , 2), B(1 , 3) and C(–4 , k) are given. If AB ⊥ BC, find the value
of k.
A.
7
B.
–7
C.
13
D.
–13
10. If the line segment joining A(–4 , 3) and B(4 , –1) cuts the x-axis at P, find AP :
PB.
A. 1 : 1
B.
2:1
C.
3:1
D.
3:2
11. Find the equation of the straight line passing through (2 , 3) and with x-intercept
5.
A.
C.
x+y+5=0
x–y–1=0
B.
D.
2
x+y–5=0
x–y+1=0
20181122 F.6 MC Practice (Coordinate Geometry)
12.
In the figure, find the equation of the straight line L.
A.
5x – 4y + 12 = 0
B.
5x + 4y – 12 = 0
C.
4x + 5y – 15 = 0
D.
4x – 5y + 15 = 0
13. The inclination of the straight line L is 45° and L passes though (4 , 8). Find the
equation of L.
A.
x – y + 4 = 0 B.
x–y–4=0
C.
x – y + 8 = 0 D.
x + y – 12 = 0
14. Which of the following straight lines is parallel to the straight line L:
A.
3x – 4y + 6 = 0
B.
3x + 4y – 8 = 0
C.
4x – 3y + 5 = 0
D.
4x + 3y – 10 = 0
x y
+ =2?
3 4
15. The x-intercept of the straight line L1: ax + 8y + 16 = 0 is –8. The straight line L2:
3x + by – 6 = 0 is parallel to L1. Find the values of a and b.
A.
a = 1, b = –24
B.
a = 1, b = 24
C.
a = 2, b = –12
D.
a = 2, b = 12
16.
In the figure, the equation of the straight line L1 is 6x – 7y + 24 = 0. The straight
line L2 passes through (–5 , 7) and (11 , 5). Find the coordinates of the point of
intersection of L1 and L2.
A.
(3 , 6)
B.
(4 , 6)
C.
3
(6 , 3)
D.
(8 , 6)
20181122 F.6 MC Practice (Coordinate Geometry)
17. Which of the following straight lines intersects the straight line
L: 2x + 5y – 30 = 0 at infinitely many points?
A.
2x – 5y = 0
B.
5x – 2y + 12 = 0
C.
4x + 10y – 36 = 0
D.
x y
+ =1
15 6
18. The circle with centre at (5 , 3) passes through (8 , 0). Find the equation of the
circle.
A.
x2 + y2 – 6x – 10y + 16 = 0
B.
x2 + y2 – 6x – 10y + 18 = 0
C.
x2 + y2 – 10x – 6y + 16 = 0
D.
x2 + y2 – 10x – 6y + 34 = 0
19. Find the coordinates of the centre and the radius of the circle
4x2 + 4y2 + 24x – 40y + 111 = 0.
A.
Coordinates of the centre = (–3 , 5), radius =
5
2
B.
Coordinates of the centre = (3 , –5), radius =
5
2
C.
Coordinates of the centre = (–12 , 20), radius = 433
D.
Coordinates of the centre = (12 , –20), radius = 433
20. The equation of circle C is x2 + y2 + 2x – 8y – 8 = 0. Find the area of C.
A.
3π
B.
5π
C.
4
9π
D.
25π
20181122 F.6 MC Practice (Coordinate Geometry)
Level 2
21.
Refer to the figure. Find the polar coordinates of P and ∠XOP.
Polar Coordinates of P
∠XOP
A.
(4 , 210°)
150°
B.
(4 , 210°)
210°
C.
(5 , 210°)
150°
D.
(5 , 210°)
210°
22.
Refer to the figure. Which of the following angles is the smallest?
A. ∠JOL
B.
C.
∠KOM
5
∠LON
D.
∠MOJ
20181122 F.6 MC Practice (Coordinate Geometry)
23.
Given that S(6 , 110°), T(2 , 200°), U(4 , 290°) and V(5 , 200°) are four points in a
polar coordinate plane, find the area of quadrilateral STUV.
A. 9 sq. units
B.
12 sq. units
C.
15 sq. units
D.
18 sq. units
24. E(3 , 20°) and F(5 , 300°) are two points in a polar coordinate plane.
Find !EOF.
A. 320°
B.
300°
C.
120°
D.
80°
25.
L3
L2
O
L1
In the figure, the slopes of the straight lines L1, L2 and L3 are m1, m2 and m3
respectively. Which of the following must be true?
A. m1 < m2 < m3
B.
m2 < m1 < m3 C.
6
m3 < m1 < m2 D.
m3 < m2 < m1
20181122 F.6 MC Practice (Coordinate Geometry)
26.
y
P(a , b)
Q(c , 0)
O
x
In the figure, O(0 , 0), P(a , b) and Q(c , 0) are the vertices of a triangle. It is
given that the slopes of OP and PQ are m and −m respectively. Which of the
following must be true?
I. c = 2a
A. II only
II.
B.
OP = PQ
III only
III. ∠OPQ = 90°
C. I and II only D.
I, II and III
27. The straight line 13x + 5y – 65 = 0 cuts the x-axis and the y-axis at A and B
respectively. If O is the origin, find the area of !OAB.
A.
13
2
B.
65
2
C.
13
D.
65
28. The straight line ℓ : 5x – 6y + 30 = 0 cuts the x-axis and the y-axis at A and B
respectively. Find the equation of the perpendicular bisector L of AB.
A.
6x + 5y + 11 = 0
B.
6x – 5y + 11 = 0
C.
12x + 10y + 11 = 0
D.
12x – 10y + 11 = 0
29. The straight line L1: ax + 9y + 6 = 0 is perpendicular to the straight line
L2: 3x – 5y – 6 = 0, where a is a non-zero real number. Find the value of a.
A.
–15
B.
–3
C.
3
D.
15
30. If a > 0, b < 0 and c > 0, which of the following graphs represents the straight
line bx + cy = a?
A.
B.
C.
D.
7
20181122 F.6 MC Practice (Coordinate Geometry)
31.
The figure shows the graph of the straight line L: ax + y + b = 0. Which of the
following are true?
I.
a>0
II.
b<0
III. a + b > 0
A.
I and II only
B.
I and III only
C.
II and III only
D.
I, II and III
32.
In the figure, the equations of the straight lines L1 and L2 are x + ay = b and
cx + y = d respectively. Which of the following are true?
I.
a>0
II.
ac > 1
III. b > ad
A.
I, II and III only
B.
I, II and IV only
C.
I, III and IV only
D.
II, III and IV only
8
IV. d > bc
20181122 F.6 MC Practice (Coordinate Geometry)
33.
In the figure, the y-intercepts of the straight lines L1 and L2 are both 4, and the xintercepts of the straight lines L1 and L3 are both 1. If L2 is parallel to L3, which
of the following is/are true?
1
.
4
I.
The slope of L1 is
II.
L2 and L3 have no points of intersection.
III. The point (–1 , 10) lies on L1.
A.
I only
B.
II only
C.
III only
D.
I and III only
34. The coordinates of A are (3 , –9). A is reflected in the x-axis to B. B is then
rotated clockwise about the origin through 90° to C. Find the equation of BC.
A.
x – y + 12 = 0
B.
x+y+6=0
C.
2x – y + 3 = 0
D.
2x + y – 15 = 0
35. The straight line L: x − 3y + 2a = 0 passes through the centre of the circle
x2 + y2 + 2x − 10y + 17 = 0. Find the value of a.
A. 8
B. 9
C. 10
D.
11
36. If the straight line 2x + 11y + k = 0 divides the circle x2 + y2 + 5x – 2y – 6 = 0
into two equal parts, find the value of k.
A.
–8
B.
–6
C.
6
D.
12
37. The circle with centre at G(−3 , 1) intersects the x-axis at two points A and B,
where AB = 2. Find the equation of the circle.
A. x2 + y2 − 6x + 2y + 8 = 0
B.
x2 + y2 + 6x − 2y − 10 = 0
C. x2 + y2 − 6x + 2y − 10 = 0
D.
x2 + y2 + 6x − 2y + 8 = 0
9
20181122 F.6 MC Practice (Coordinate Geometry)
38.
y
G
A
O B
x
In the figure, the circle with centre at G touches the y-axis at A(0 , 4). B(2 , 0) is
one of the points of intersection of the circle and the x-axis. Find the equation of
the circle.
A.
x2 + y2 − 8y + 9 = 0
B.
x2 + y2 − 2x − 4y = 0
C.
x2 + y2 − 6x − 8y + 10 = 0
D.
x2 + y2 − 10x − 8y + 16 = 0
39. If L is the tangent to the circle x2 + y2 − 10x + 6y + 32 = 0 at E(4 , −2), find the
equation of L.
A.
x − y − 6 = 0 B.
x + y − 2 = 0 C.
x + 2y − 1 = 0 D.
2x + 2y − 5 = 0
40. The equation of a circle is x2 + y2 + 6x – 2y + 5 = 0. Which of the following are
true?
I.
The straight line y = 3x + 10 passes through the centre of the circle.
II.
The origin lies outside the circle.
III. There is only 1 point of intersection of the circle and the x-axis.
A.
I and II only
B.
I and III only
C.
II and III only
D.
I, II and III
10
20181122 F.6 MC Practice (Coordinate Geometry)
Level 3
41. If a < 0 and b > 0, which of the following graphs represents the straight line
x y
+ = −1 ?
a b
A.
B.
C.
D.
42.
In the figure, ABCD is a square. The equations of AB and AD are
2x + 7y – 29 = 0 and 7x – 2y + 31 = 0 respectively. The coordinates of B are
(4 , 3). Find the area of square ABCD.
A.
34
B.
53
C.
65
D.
113
43.
In the figure, the straight lines L1: y = ax + b and L2: y = cx + d intersect at a
point on the positive y-axis. Which of the following must be true?
A.
ab > 0
B.
cd < 0
C.
11
b=d
D.
ad = bc
20181122 F.6 MC Practice (Coordinate Geometry)
44. If a < 0, which of the following shows the graph of the straight line
ax + 3y – 5 = 0?
A.
B.
y
y
x
O
x
O
C.
D.
y
y
x
O
x
O
45.
y
L
P
R
Q
x
O
In the figure, the equation of the straight line L is 2x – y + 4 = 0. Find the length
of OR.
A.
5
B.
4 5
5
C.
2 5
D.
3 5
46. If the x-intercept of a diameter of the circle x2 + y2 + kx – 14y – 11 = 0 is –11 and
7
the slope of the diameter is , find the value of k.
6
A. –10
B. –5
C. 5
D. 10
47. The equation of a circle is x2 + y2 − 8x + 4y + 15 = 0. Which of the following are
true?
I.
The circle is a real circle.
II. The circle passes through point (6 , −1).
III. The straight line y = 11 – 2x does not intersect the circle.
A.
I and II only
B.
I and III only
C.
II and III only
D.
I, II and III
12
20181122 F.6 MC Practice (Coordinate Geometry)
48. The equation of a circle with centre G is x2 + y2 − 18x + 10y + 97 = 0. A and B are
two points on the circle. It is given that the perpendicular distance from G to AB
is 2 2 . Find the ratio of the radius to the length of AB.
A. 2 : 1
B. 3 : 1
C. 3 : 2
D. 4 : 3
49.
y
A(0 , 3)
B
C
O
x
In the figure, the circle passes through A(0 , 3). The circle intersects the x-axis at
two points B and C. !ABC is an equilateral triangle. Find the equation of the
circle.
A.
C.
x2 + y2 − 2 3 y + 2 = 0
x2 + y2 − 2y − 2 = 0
B.
D.
x2 + y2 − 3 y + 3 3 − 9 = 0
x2 + y2 − 2y − 3 = 0
50.
y
P
G
Q
O
x
R
In the figure, the circle with centre at G touches the x-axis at R and intersects the
y-axis at P(0 , 25) and Q(0 , 1). Find the equation of the circle.
A.
x2 + y2 − 10x + 25y = 0
B.
x2 + y2 − 10x − 26y + 25 = 0
C.
x2 + y2 − 5x − 26y + 25 = 0
D.
x2 + y2 − 5x − 25y − 10 = 0
13
20181122 F.6 MC Practice (Coordinate Geometry)
51.
y
A(0 , 1)
B(–3 , 0)
x
O
In the figure, the centre G of the circle lies on the x-axis. The circle intersects the
negative x-axis at B(–3 , 0) and intersects the positive y-axis at A(0 , 1). Find the
equation of the circle.
2
2
A.
4
5

2
x+  + y =
3
3

C.
4
5

x2 +  y +  =
3
3

B.
4
25

2
x+  + y =
3
9

D.
4
25

x2 +  y +  =
3
9

2
2
52. A circle lies in quadrant I and its centre lies on the angle bisector of the x-axis
and the y-axis. The distance between the centre and the x-axis is 3"#The distance
between the centre and the origin is 3 times the radius. Find the equation of the
circle.
A.
x2 + y2 − 6x − 6y + 16 = 0
B.
x2 + y2 − 6x − 4y + 10 = 0
C.
x2 + y2 − 6x + 4 = 0
D.
x2 + y2 − 6x − 6y − 9 = 0
53. If the straight line x + 2y = k intersects the circle x2 + y2 − 9x − 4y + 5 = 0 at two
points A and B, find the x-coordinate of the mid-point of AB.
A.
k −7
5
B.
k + 14
5
C.
14
2k − 14
5
D.
2k + 28
5
20181122 F.6 MC Practice (Coordinate Geometry)
54.
y
L
C
B(–7 , 12)
A
x
O
In the figure, the straight line L passes through B(−7 , 12) and touches the circle
C: x2 + y2 − 6x − 14y + 13 = 0 at A. Find the length of AB.
A.
45
B.
80
C.
10
D.
125
55.
y
B
C
θ
x
A
O
In the figure, C is the centre of the circle x2 + y2 – 24x – 18y + 144 = 0. OA and
OB touch the circle at A and B respectively, where A lies on the x-axis.
If ∠AOB = θ, find the value of sin
A.
1
5
B.
θ
2
.
2
5
C.
3
5
D.
4
5
x y
+ = 1 touches the circle
a b
(x − a)2 + (y − b)2 = k, where a and b are non-zero constants and k > 0. Express k
in terms of a and b.
a 2b 2
A. a2 + b2
B.
C. a2b2
D. (a + b)2
a 2 + b2
56. It is given that the straight line
15
20181122 F.6 MC Practice (Coordinate Geometry)
57. The equation of the circle in the figure is
(x – h)2 + (y + k)2 = r2, where h, k and r are positive
constants. Which of the following are true?
I.
h–k>0
II.
r–h<0
III. r – k > 0
A.
I and II only
B.
I and III only
C.
II and III only D.
I, II and III
58. The equation of a circle is (x – a)2 + (y + b)2 = (a + b)2, where a > 0 and b > 0.
Which of the following are true?
I.
The circle passes through the origin.
II.
The coordinates of the centre of the circle are (a , –b).
III. (–b , 0) lies outside the circle.
A.
I and II only
B.
I and III only C.
II and III only D.
I, II and III
59. The circle x2 + y2 – 10x + 5y + 20 = 0 and the straight line y = mx intersect at two
distinct points. Find the range of values of m.
2
11
A.
−2< m<
C.
m < –2 or m >
2
11
2
<m<2
11
B.
−
D.
m< −
2
or m > 2
11
60. In the figure, A, B and C are the vertices of
!ABC. Which of the following is the equation of
the circumcircle of !ABC?
A.
5x2 + 5y2 – 12x + 5y + 74 = 0
B.
5x2 + 5y2 – 12x + 74 = 0
C.
5x2 + 5y2 – 24x – 148 = 0
D.
5x2 + 5y2 – 24x + 5y + 148 = 0
Full Solution:
16