Analytical Geometry of the Straight Line

MATHEMATICS

Learner’s Study and

Revision Guide for

Grade 12

STRAIGHT LINE

Revision Notes, Exercises and Solution Hints by

Roseinnes Phahle

Examination Questions by the Department of Basic Education

Preparation for the Mathematics examination brought to you by Kagiso Trust

2

Contents

Unit 13

All you need to know: Revision notes 3

Exercise 13 4

Answers 5

Examination questions with solution hints and answers 7

More questions from past examination papers 12

Answers 19

How to use this revision and study guide

1. Study the revision notes given at the beginning. The notes are interactive in that in some parts you are required to make a response based on your prior learning of the topic from your teacher in class or from a textbook. Furthermore, the notes cover all the Mathematics from Grade 10 to Grade 12.

2. “Warm‐up” exercises follow the notes. Some exercises carry solution HINTS in the answer section. Do not read the answer or hints until you have tried to work out a question and are having difficulty.

3. The notes and exercises are followed by questions from past examination papers.

4. The examination questions are followed by blank spaces or boxes inside a table. Do the working out of the question inside these spaces or boxes.

5. Alongside the blank boxes are HINTS in case you have difficulty solving a part of the question. Do not read the hints until you have tried to work out the question and are having difficulty.

6. What follows next are more questions taken from past examination papers.

7. Answers to the extra past examination questions appear at the end. Some answers carry HINTS and notes to enrich your knowledge.

8. Finally, don’t be a loner. Work through this guide in a team with your classmates.

Analytical geometry of the straight line

REVISION UNIT 13: ANALYTICAL GEOMETRY

All you need to know: What follows below is all you need to know in order to answer the

question on the analytical geometry of the straight.

Consider two points A ( )11 , yx and B ( )22 , yx .

1. Distance between A and B is ( ) ( ) ][ 212

212 yyxx −+−

2. Midpoint of the line joining A and B has coordinates ⎟⎠⎞

⎜⎝⎛ ++

2;

22121 yyxx

3. The gradient of the line joining A and B is 12

12

xxyy

m−−

=

4. If two lines are parallel, they have equal gradients.

5. If two lines are perpendicular, the product of their gradients is ‐1.

6. All straight line equations can be written in the form cmxy += where m is the gradient and

( )c;0 is the point at which the line cuts the y ‐ axis or the y ‐intercept.

7. The gradient of a line can also be measured by the tangent of the angle θ between the positive direction of the x ‐axis and the line, measured anti‐clockwise. That is:

θtan=m

8. The equation of the line joining the points A and B is given by:

( )21 xxmyy −=− or ( )112

121 . xx

xxyy

yy −−−

=− or 12

12

1

1

xxyy

xxyy

−−

=−−

9. Intersection of lines: In order to find the point in which two lines intersect we have to find a point with coordinates which satisfy both equations. We find this point by solving the equations of the lines simultaneously.

10. Intersection of a straight line and a curve: A straight line may intersect a curve at more than one point. Thus solving the equations of the line and the curve simultaneously could give more than one answer, these being the points in which the line and curve intersect.

11. Collinearity: Points A, B and C are collinear if they are joined by lines of equal slopes.


4

EXERCISE 13

13.1. A triangle has vertices at A(0;8), B(1;1) and C(5;3). Depict this triangle in a sketch. Show that the triangle is isosceles and find:

13.1.1 the equation of the straight line through A and C,

13.1.2 the coordinates of the foot of the perpendicular from B to AC,

13.1.3 the length of the perpendicular from B to AC.

13.2 ` Find the coordinates of the vertices of the triangle whose sides are given by the equations 72 =− xy , 1952 +−= xy and 332 =+ xy . Illustrate this question with a diagram.

13.3 Determine

13.3.1 the equation of the straight line passing through the point (8;‐2) and which is at right angles to the line 72 += xy ,

13.3.2 the coordinates of the point where this line intersects the line 72 += xy ,

13.3.3 the distance of the point (8;‐2) from the line 72 += xy .

13.4. Determine

13.4.1 the equation of the perpendicular from (7;3) to the line 26 =− yx ,

13.4.2 the length of the perpendicular from (7;3) to the line 26 =− yx .

13.5 The line 42 += xy cuts the curve xxy 32 −= at the points A and B.

13.5.1 What are the coordinates of A and B?

13.5.2 What are the coordinates of the midpoint of AB?

13.5.3 What is the equation of the line perpendicular to AB and passing through the midpoint of AB?

13.5.4 Illustrate your answer with a sketch.

13.6 The curve xxy 22 += is met by the line 63 += xy at two points A and B.

13.6.1 Calculate the coordinates of A and B.

13.6.2 Find the equation of the line joining A and B.

13.6.3 Find the length of AB.


ANSWERS

EXERCISE 13

-13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12 13

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

x

y(0,8)

(1,1)

(5,3)

A

B

C(4,4)

13.1.1 y=-x+8

13.1.2 HINT: First find the equation of the perpendicular. Thensolve its equation simultaneously with the equation of AC to findthe coordinates of its foot on AC.Answer: (4;4)13.1.3 3sq root 3

13.2 The coordinates are (‐1;3), (2;4,5) and (8;‐9,5). Illustration:

f(x)=0.5(x+7)

f(x)=0.5(-5x+19)

f(x)=0.5(-3x+3)

-4 -3 -2 -1 1 2 3 4 5 6 7 8

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

2y-x=72y=-5x+19

2y+3=3

Coordinates of the vertices are:(-1;3), (2;-4,5) and (8;-9,5)


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13.3.1 42 +−= xy 13.3.2 (‐2;3)

13.3.3 55

13.4.1 256 +−= xy

13.4.2 37

13.5.1 Coordinates are A(‐0,7;2,6) and B(5,7;15,4) 13.5.2 Coordinates of midpoint of AB are (2,5; 9) 13.5.3 Equation of line perpendicular to AB and passing through the midpoint of AB is

232 +−= xy 13.5.4 Sketch:

-13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

x

y

A

B

Line perpenicular to AB and passing throughmidpoint of Ab

13.6.1 The coordinates are A(‐2;0) and B(3,15) 13.6.2 ( )23 += xy

13.6.3 Length of AB = 105


PAPER 2 QUESTION 1 DoE/ADDITIONAL EXEMPLAR 2008


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PAPER 2 QUESTION 1 DoE/ADDITIONAL EXEMPLAR 2008

Number Hints and answers Work out the solutions in the boxes below 1.1 Use the distance formula.

Answer: AC= 208

1.2 Use the midpoint formula. Answer: M(‐1; 3)

1.3 Use the gradient formula.

Answer: 23

=ACm

1.4 There are two formulae on the formula sheet. Choose either one. Recall that the product of the slopes of lines that are perpendicular is ‐1.

Answer: 32

−=y

1.5 Area of a triangle is

21base x height.

The diagram shows the height and which side is base. Find their lengths and substitute into the formula for area. Answer: Area of ABCΔ =52 sq units

1.6 Find a way of using the fact that the tangent of the angle which a line makes with the positive direction of the x ‐axis is equal to the slope of the line. Or, you could look at ABNΔ . Answer:

o7,33≈θ


PAPER 2 QUESTION 1 DoE/NOVEMBER 2008


10

PAPER 2 QUESTION 1 DoE/NOVEMBER 2008

Number Hints and answers Work out the solutions in the boxes below 1.1 Use the midpoint formula.

Answer: ⎟⎠⎞

⎜⎝⎛ −−

21;

21M

1.2 Find the midpoint of BD and see if it is the same as the midpoint of AC you found in 1.1. Or, show that AC and BD bisect each other. Answer: Write down your conclusion.

1.3 Use the product of gradients of perpendicular lines is equal to ‐1. That is, find the gradients of AD and AC and see if their product is ‐1. There are other ways of proving that

o90CD̂A = . Can you find these other ways? Answer: Write down your conclusion.


Number Hints and answers Work out the solutions in the boxes below 1.4 There is more than one way to answer

this question. One way: Using your knowledge of the properties of the square:

1. Show using the distance formula that the diagonals are equal in length;

2. Use what you proved in 1.2 that the diagonals bisect each other;

3. Use 1.3 that they are right angles.

Can you find other ways of showing that ABCD is a square? Answer: Write down your conclusion.

1.5 Use m=θtan to find the size of θ . This means that you must m the gradient of DC.

Answer: o7,123=θ

1.6 Is the length of OC equal to radius 2? Find out. Answer: Write down your conclusion.


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MORE QUESTIONS FROM PAST EXAMINATION PAPERS

Exemplar 2008


Preparatory Examination 2008


14

Feb – March 2009


November 2009 (Unused)


16

November 2009 (1)


Feb – March 2010


18

Feb – March 2010


ANSWERS

Exemplar 2008

1.1 AC = 102 1.2 M(‐1; 4) 1.3 Proof required. Provide a proof and check with the teacher if it is correct. 1.4 Area Δ ABC = 20 1.5 1−−= xy

1.6 o135=θ

1.7 o53,1CB̂A = Preparatory Examination 2008

1.1.1 BC = 262 1.1.2 M(1; ‐1)

1.1.3 5BC =m

1.1.4 o69,78=θ

1.2 54

51

−−= xy

1.3 A(11; ‐3) 1.4 C’(0; ‐12)

1.5 14

ABC AreaC'B'A' Area

=ΔΔ

Feb/March 2009

1.1 31

BC =m

1.2 3

1731

+= xy

1.3 8=t

1.4 AB = 102 1.5 Proof required. Provide a proof and check with the teacher if it is correct. 1.6 Area of ABCD = 30 sq units

o43,18=θ

November 2009 (Unused paper)

1.1 1AC =m

1.2 4−= xy 1.3 Proof required. Provide a proof and check with the teacher if it is correct. 1.4 Proof required. Provide a proof and check with the teacher if it is correct. 1.5 Area Δ ABC = 36 sq units November 2009(1)

4.1 t-1

3or 1AB =m

4.2 t = ‐2 4.3 Midpoint of BC = (0; ‐20 4.5 6−= xy Feb/March 2010 4.1 Proof required. Provide a proof and check with the teacher if it is correct. 4.2 A(2; 1) 4.3 32 −= xy 4.4 BQ = 5 4.5 Proof required. Provide a proof and check with the teacher if it is correct. 4.6 R(4; 5)

5.1 4CD =m

5.2 164 −= xy

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Analytical Geometry of the Straight Line