SPM Add Math Form 4 Chapter 6 Coordinate Geometry

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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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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 3

2. Find the x-intercept of

the line.

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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 2

1 also passes through the point (6,k),

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.

27) The x-intercept and the y-intercept of a straight line are 5 and -3 respectively. Find the

equation of the 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

(a) The equation of the line PQ using intercepts,

(b) The gradient of the line PQ.

29) The line joining the points A(5,0) and B(0,-12) passes through the point (10,p). Find

(c) The value of p,

(d) The distance of AB.

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 2

3 ,

find the equation of CD and

hence the coordinates of C.

31) Find the intercepts and gradient of the line 143

yx

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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The equation of a straight line in general form

33) A straight line of gradient 3

2 passing through the point (-1,5). Find the equation of the line

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 .352

yx

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.

36) The diagram shows a quadrilateral whose vertices are A(4,0), B(4,4), C(-2,3) and D(0,-2).

Find

(a) The equations of its diagonals,

(b) The coordinates of the point of

intersection of the diagonals.

Parallel and Perpendicular Lines

The parallel lines

37) Given that the line 6x+3y = 2 is parallel to the line 52

xp

y , find the value of p.

38) Show that the points A(4,1), B(8,2), C(5,4) and D(1,3) are the vertices of a parallelogram.

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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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.

AB : y=hx-k

PQ : y=(2k-3)x +4

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.

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.

47) Two points have coordinates A(-2,5) and B(6,3). Find

(d) the midpoint of AB,

(e) the equation of the perpendicular bisector of AB.

48) The diagram shows a straight line AB with the equation .153

yx

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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(a) the coordinates of P and S,

(b) the equation of QR,

(c) the coordinates of R,

(d) 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°.

Find

(a) The equations of PS and RS,

(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.

(a) Write down the equation of AB

in the form of intercepts.

(b) Given that 2AD=DB, find the

coordinates of D.

(c) Given that CD is perpendicular

to AB, find the y-intercept

of CD.

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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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.

58) A point P moves so that its distance from Q (2,0) is equal to its distance from the line x=-2.

Find the equation of the locus of P.

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 A(4,-3) and B(-2,3) are two fixed points. Point P moves in such a way that the ratio

of PA to PB is 2:1.

(a) Show that the equation of the locus of P is x2 + y2 +8x -10y+9 = 0.

(b) 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.

61) A point P moves along the arc of a circle with centre A(3,2). The arc passes through

Q(0,-2) and R(6,k).

(a) Find (i) the equation of the locus of P,

(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.