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. Copyright www.epitomeofsuccess.com

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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.

For the Complete Worksheet and Answers, please join as a member 27) The x-intercept and the y-intercept of a straight line are 5 and -3 respectively. Find the atof the www.epitomeonlinetuition.com. 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

When you join as a Bronze or Silver Member, you will gain unlimited access to all the Worksheets and Video for SPM Add Math 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. Copyright www.epitomeofsuccess.com

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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.

For the Complete Worksheet and Answers, please join as a member at (a) The equations of its diagonals, www.epitomeonlinetuition.com. (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.

When you join as a Bronze or Silver Parallel and Perpendicular Lines Member, you will gain unlimited The parallel lines p access all the Worksheets 37) Given that the line 6x+3yto = 2 is parallel to the line y  x  5 , find the and value of p. 2 Video Tutorials for SPM Add Math 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.

For the Complete Worksheet and PQ : y=(2k-3)x +4 Answers, please join as a member at www.epitomeonlinetuition.com. 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.

When you join as a Bronze or Silver Member, you will gain unlimited access to all the Worksheets and 47) Two points have coordinates A(-2,5) and B(6,3). Find Video Tutorials for SPM Add Math (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°.

For the Complete Worksheet and Find Answers, please join as a member (a) The equations of PS and RS, at www.epitomeonlinetuition.com. (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.

When you join as a Bronze or Silver (a) Write down the equation of AB Member, you will gain unlimited in the form of intercepts. (b)access Given that 2AD=DB, to find allthethe Worksheets and 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. Copyright www.epitomeofsuccess.com

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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.

For the Complete Worksheet and 58) A point PAnswers, moves so that its distance from Q (2,0) is equal to its distance from the lineat x=-2. please join as a member Find the equation of the locus of P. www.epitomeonlinetuition.com. 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.

When you join as a Bronze or Silver Member, you will gain unlimited access to all the Worksheets and Video Tutorials 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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