SPM Add Math Form 4 Chapter 6 Coordinate Geometry
CHAPTER 6 : COORDINATE GEOMETRY
Distance between Two Points
Formula for distance between two points
1) Find the distance between two points P(-1,1) and Q(2,5).
2) The distance between two points A(3,-1) and B(-1,m) is 5 units. Find the values of m.
3) The diagram shows a triangle ABC.
The coordinates of A,B and C are (-2, -6),
(-4,8) and (4,2) respectively.
Find
(a) the length of AC and BC and show that
the triangle is an isosceles triangle,
(b) the perimeter of the triangle ABC.
Division of a Line Segment
The midpoint of two given points
4) Find the coordinates of the midpoint of the line segment joining P(2,-5) and Q(4,7).
5) If M(7,-3) is the midpoint of the line joining A(5,-4) and B(p,q), find the value of p and q.
6) Three of the vertices of a parallelogram ABCD are A(1,2) , B(3,5) and D(2,7). Find
(a) the coordinates of the midpoint BD,
(b) the coordinates of C.
Point that dividing a line segment in the ratio m : n
7) Find the coordinates of the point P that divides the line segment joining 2 points A(0,-2) and
B(5,8), in the ratio 3:2.
8) P is a point on the line segment joining A(3,1) and B(-3,4) such that AP = 2PB. Find the
coordinates of P.
9) A line segment has end points at A(-6,1) and B(x,y). If P(-2,3) divides AB internally in the
ratio 2:3, find the coordinates of B.
10) The points P(1,k), A(3,-1) and B(-2,4) lie on the straight line such that AP : PB = m:n. Find
(a) the ration of m:n,
(b) the value of k.
11) The points A(3h,h) , B(p,q) and C(2p,4q) lie on a straight line. B divides AC internally in the
ratio 1:2. Express p in terms of q.
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SPM Add Math Form 4 Chapter 6 Coordinate Geometry
Areas of Polygons
Formula for area of triangle
12) Find the area of triangle ABC where A, B and C are (2,3), (5,1) and (3,7) respectively.
13) The vertices of a triangle are A(2,3) , B(2k,k) and C(0,-1). If the area of the triangle ABC is
4 unit2, find the positive values of k.
14) Show that the points A(-2,-1) , B(0,1) and C(3,4) are collinear.
15) Given that the points A(1,5) , B(5,k) and C(9,1) are collinear, find the value of k.
Formula for area of quadrilateral
16) ABCD is a rhombus with coordinates (4,3), (p,3), (-4,-1) and (1,q) respectively. Find
(a) the value of p and q.
(b) the area of the rhombus.
Equations of Straight Lines
The gradient of a straight line
17) Find the gradient of the line passing through each of the following pairs of points.
(a) A(2,4) and B (4,8)
(b) C(2,1) and D(1,4)
(c) E(3,5) and F(7,5)
(d) G(6,-4) and H(6,3)
18) The line joining the points A(2,-3) and B(3,k) has gradient -3. Find the value of k.
19) Given that the points P(t,5) , Q(3,4) and R(9,t) are collinear, find the positive values of t.
The gradient of a straight line using the intercept
20) Find the gradient of the following lines.
(a)
(b)
21) Find the gradient of the straight line that passes through the points (0,14) and (-7,0).
22) A straight line passing through the point (0,-6) has a gradient of
2
. Find the x-intercept of
3
the line.
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SPM Add Math Form 4 Chapter 6 Coordinate Geometry
The equation of a straight line
23) Find the equation of the line that passing through (5,2) and has gradient -3.
24) If the line through the point A(4,-5) with gradient
1
also passes through the point (6,k),
2
find the value of k.
25) Find the equation of the line passing through (3,-2) and (-4,5).
26) A line of gradient -2 passing through A(-1,4) meets the x-axis at B. Another line through A
meets the x-axis at C(-2,0).
(a) Find the equation of AB and of AC.
(b) Calculate the area of ΔABC.
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27) The x-intercept and the y-intercept of a straight line are 5 and -3 respectively. Find the
atof the
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equation
straight line.
28) The diagram shows a straight line crosses the x-axis at at point P(3,0) and the y-axis at
Q(0,4). Find
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29) The line
joining the Tutorials
points A(5,0) and B(0,-12)
passes through
the point
(10,p). Find
(c) The value of p,
both Form 4 and Form 5
(d) The distance of AB.
(a) The equation of the line PQ using intercepts,
(b) The gradient of the line PQ.
30) The diagram shows a straight line CD which meets a straight line AB at the point D.
The point C lies on the y-axis.
(a) Write down the equation of AB
by using intercepts.
(b) Given that AD:DB = 1:2, find the
coordinates of D.
(c) Given that the gradient of CD is 
3
,
2
find the equation of CD and
hence the coordinates of C.
31) Find the intercepts and gradient of the line
x y
 1
3 4
32) Express the equation of the straight line 4x+3y=12 in intercept form. Hence, state the
intercepts and the gradient of the line.
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SPM Add Math Form 4 Chapter 6 Coordinate Geometry
The equation of a straight line in general form
33) A straight line of gradient
2
passing through the point (-1,5). Find the equation of the line
3
and express it in general form.
The point of intersection of two straight lines
34) Find the coordinates of the point of intersection of the lines 2x+3y = -7 and
x y
  3.
2 5
35) The straight line passing through the points (-2,5) and (2,1) intercepts the line 3y-x=5 at the
point P. Find the coordinates of P.
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(a) The equations of its diagonals,
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(b) The coordinates of the point of
36) The diagram shows a quadrilateral whose vertices are A(4,0), B(4,4), C(-2,3) and D(0,-2).
Find
intersection of the diagonals.
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The parallel
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37) Given that
the line 6x+3yto
= 2 is
parallel
to the
line y  x  5 , find the and
value of p.
2
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38) Show that the points A(4,1), B(8,2), C(5,4) and D(1,3) are the vertices of a parallelogram.
both Form 4 and Form 5
The equation of a straight line parallel to a given line
39) Find the equation of the line passing through the point A(4,7) and parallel to the line y=3x-4.
40) The diagram shows a parallelogram PQRS in which P is (7,1). The equations of QR and
RS are y-x=2 and 3y+x=-6 respectively. Find
(a) The coordinates of R,
(b) The equations of PQ and PS.
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SPM Add Math Form 4 Chapter 6 Coordinate Geometry
Perpendicular lines
41) Determine whether the lines y-4x+7=0 and 4y+x=8 are perpendicular.
42) The line y=ax+7 is perpendicular to the line y=2x-3. State the value of a.
43) Show that the points P(1,5), Q(-1,1), and R(3,-1) are the vertices of a right-angled triangle.
44) The following information refers to the equations of two straight lines AB and PQ which are
perpendicular to each other. Express h in terms of k.
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PQ : y=(2k-3)x +4
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AB : y=hx-k
The equation of a straight line perpendicular to a given line
45) Find the equation of the line passing through the point (-3,2) and perpendicular to the line
y+2x=1.
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47) Two points
have coordinates
A(-2,5) and
B(6,3).
Find
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SPM
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(d) the midpoint of AB,
(e) the equation
of the perpendicular
both
Form 4bisector
andof AB.
Form 5
46) Three points have coordinate A(4,4), B(-2,1) and C(3,1). Find
(a) the equation of the line through A and perpendicular to AB,
(b) the equation of the line through C and parallel to AB,
(c) the coordinates of the point D at which these two lines intersect.
48) The diagram shows a straight line AB with the equation
x y
  1.
3 5
The points A and B lie on the x-axis
and y-axis respectively.
Find the equation of the straight line
perpendicular to AB and passing through
the point A.
Problems involving equations of straight lines
49) The diagram shows a quadrilateral PQRS in which P is on the x-axis and S is on the y-axis.
Q is the point (8,3) and the equation of RS is y=2x+2. PS and QR are perpendicular to RS.
Find
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SPM Add Math Form 4 Chapter 6 Coordinate Geometry
(a)
(b)
(c)
(d)
the coordinates of P and S,
the equation of QR,
the coordinates of R,
the area of the quadrilateral PQRS.
50) In the diagram, the points P, Q and R are (-1,2), (1,-1) and (4,1) respectively. The line PS is
parallel to QR and angle PRS = 90°.
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(a) The equations of PS and RS,
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(b) The coordinates of S.
51) The diagram shows a straight line CD which meets a straight line AB at the point D. The
point C lies on the y-axis.
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(a) Write down the equation of AB
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Given that 2AD=DB,
to find
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coordinates of D.
(c)Video
Given that CD
is perpendicular for SPM Add Math
Tutorials
to AB, find the y-intercept
of CD. both Form 4 and Form 5
52) In the diagram below, the line AB is perpendicular to the line BC. The coordinates of A is
(-4,3) and the equation of BC is 3y-x+7=0. Find
(a) (i) the equation of the straight line AB,
(ii) the coordinates of B,
(b) The straight line AB is extended to a
point D such that AB:BD = 2:1.
Find the coordinates of D.
Equation of Locus
The equation of the locus of a moving point
53) Find the equation of the locus of a moving point P such that its distance from the point A(2,5)
is 3 units.
54) Find the equation of the locus of point P such that its distances from A(-2,4) and B(3,5) are
equal.
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SPM Add Math Form 4 Chapter 6 Coordinate Geometry
55) A point P moves such that its distances from A(0,1) and B(3,4) are in the ratio 1:2. Find
the equation of the locus of P.
56) The point A is (-2,1) and the point B is (6,4). The point P moves such that PA:PB=2:3. Find
the equation of the locus of P.
Problems involving loci
57) The points A(0,6), B(6,0) and P(x,y) lie on the circumference of a circle with diameter AB.
Find the equation of the locus of the following point P.
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58) A point PAnswers,
moves so that its distance
from Q (2,0)
is equal
to its
distance
from the lineat
x=-2.
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Find the equation of the locus of P.
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59) A circle with centre P touches the y-axis and passes through the point M(4,2). Find the
equation of the locus of P.
60) Points
of PA
(a)
(b)
A(4,-3) and B(-2,3) are two fixed points. Point P moves in such a way that the ratio
to PB is 2:1.
Show that the equation of the locus of P is x2 + y2 +8x -10y+9 = 0.
The point C(0,1) lies on the locus of P. Show that the points A, C and B lie on a
straight line and find its equation.
(c) Given that this line meets the locus of P again at point D, find the coordinates of D.
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61) A point Pfor
movesSPM
along the arc
of a circle
with centre
A(3,2). The
arc passes
Add
Math
both
Form
4through
and
Q(0,-2) and R(6,k).
(a) Find (i) the equation of the locus Form
of P,
5
(ii) the values of k.
(b) The tangent to the circle at point Q intersects the y-axis at point S. Find the area of
triangle OQS.
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