5 Equations of Straight Lines

5 Equations of Straight Lines

Quick Review

Let’s Try (p. 5.2)

1.

2. (a) Slope of L

(b) ∵

∴

3. (a) Slope of L1

Slope of L

∴ Slope of L1 ≠ slope of L

∴ L1 is not parallel to L.

(b)  ∵ L2 is perpendicular to L.

∴

4. (a) Let (x, y) be the coordinates of P.

∴ Coordinates of P

(b)  Let (x, y) be the coordinates of Q.

∴ Coordinates of Q =

Review Exercise 5 (p. 5.4)

1. (a) The coordinates of A are (3, 4).

(b) Length of AB

(c) Slope of AB

(d)

2. ∵ The slope of line segment joining P and Q is -2.

∴

3. (a) Slope of L

∵ L1 is parallel to L.

∴

(b) ∵ L2 is perpendicular to L.

∴

4. (a) Let (x, y) be the coordinates of A.

∴ The coordinates of A are (-6, 8).

(b) Let (x, y) be the coordinates of the mid-point of AM.

∴ The coordinates of the mid-point of AM are

(-4, 7).


5. (a) Let (x, y) be the coordinates of P.

∴ The coordinates of P are (2, 7).

(b) Let (x, y) be the coordinates of P.

∴ The coordinates of P are.

Inspiring Activity

Inspiring Activity 5.1 (p. 5.5)

1.

Point / A(5, 4) / B(4, 3) / C(2, 1)
x-coordinate / 5 / 4 / 2
y-coordinate / 4 / 3 / 1
Point / D(0, -1) / E(-2, -3) / F(-4, -5)
x-coordinate / 0 / -2 / -4
y-coordinate / -1 / -3 / -5

2. (a)

(b) The equation of the straight line L is.

(c) By substituting x = 1 and y = 0 into, we have

L.H.S. = 0

R.H.S. = 1 - 1 = 0

∵ L.H.S. = R.H.S.

∴ The coordinates of G satisfy the equation of L.

Classwork

Classwork (p. 5.12)

1. ∵ L1 is parallel to the x-axis and passes through (3, 6).

∴ The equation of L1 is y = 6.

∵ L2 is parallel to the x-axis and passes through (-2, -3).

∴ The equation of L2 is y = -3.

2. ∵ L3 is perpendicular to the x-axis and passes through

(-4, 6).

∴ The equation of L3 is x = -4.

∵ L4 is perpendicular to the x-axis and passes through

(8, -1).

∴ The equation of L4 is x = 8.

3. ∵ L5 passes through the origin with slope.

∴ The equation of L5 is.

∵ L6 passes through the origin with slope -3.

∴ The equation of L6 is y = -3x.

4. ∵ L7 passes through the origin and the point (-2, -3).

∴ Slope of L7

∴ The equation of L7 is.

∵ L8 passes through the origin and the point (3, -3).

∴ Slope of L8

∴ The equation of L8 is.

Classwork (p. 5.17)

(a)

∴ The equation of the straight line is.

(b)

∴ The equation of the straight line is.

(c)

∴ The equation of the straight line is.

(d)

∴ The equation of the straight line is.

Classwork (p. 5.18)

Slope / x-intercept / y-intercept
L1: 2x + y + 4 = 0
L2: 5x + 2y – 10 = 0
L3: 6x – 3y – 4 = 0
L4:– 9x + 4y – 8 = 0

Classwork (p. 5.26)

(a) Slope of L1 = 0

Slope of L2 = 0

∴ Slope of L1 = slope of L2 ……(1)

y-intercept of L1 = 2

y-intercept of L2 = -3

∴ y-intercept of L1 ≠ y-intercept of L2 ……(2)

By (1) and (2), L1 and L2 have no intersections.

(b) Both L1 and L2 are vertical. ……(3)

x-intercept of L1 = 4

x-intercept of L2

∴ x-intercept of L1 = x-intercept of L2 ……(4)

By (3) and (4), L1 and L2 have an infinite number of intersections.

(c) ∵ L1 is vertical but L2 is horizontal.

∴ L1 and L2 have one intersection.

(d) Slope of L1 = 3

Slope of L2 = -3

∴ Slope of L1 ≠ slope of L2

∴ L1 and L2 have one intersection.

(e) Slope of L1 = 2

Slope of L2 = 2

∴ Slope of L1 = slope of L2 ……(5)

y-intercept of L1 = 6

y-intercept of L2 = -6

∴ y-intercept of L1 ≠ y-intercept of L2 ……(6)

By (5) and (6), L1 and L2 have no intersections.

(f) Slope of L1 = –1

Slope of L2 = 1

∴ Slope of L1 ≠ slope of L2

∴ L1 and L2 have one intersection.

Quick Practice

Quick Practice 5.1 (p. 5.7)

(a) The equation of the straight line is

(b) The equation of the straight line is

Quick Practice 5.2 (p. 5.8)

(a) Slope of L


The equation of L is

(b) By substituting x = -2 and y =into, we have

L.H.S. =

R.H.S.

∵ L.H.S. = R.H.S.

∴ satisfies the equation.

∴ L passes through.

Quick Practice 5.3 (p. 5.8)

The equation of the straight line is .

Quick Practice 5.4 (p. 5.9)

(a)

∴ The slope is 3 and the y-intercept is –12.

(b)

∴ The slope is –2 and the y-intercept is .

Quick Practice 5.5 (p. 5.10)

The equation of the straight line is

∴

Quick Practice 5.6 (p. 5.10)

(a) ∵ x-intercept and the y-intercept of L are 6 and 3

respectively.

∴ L passes through (6, 0) and (0, 3).

The equation of L is

∴

(b) By substituting (4, k) into the equation of L, we have

Quick Practice 5.7 (p. 5.19)

(a) From the equation of L: -3x + 2y - 7 = 0, we have

(b) Let m1 and m2 be the slopes of L1 and L2 respectively.

(i) ∵ L1 // L

∴

The equation of L1 is

∴

(ii) ∵

∴

The equation of L2 is

∴

Quick Practice 5.8 (p. 5.20)

(a) ∵ x-intercept of L1 = –6

∴

(b) ∵ x-intercept of

∴ L2 passes through (6, 0).

∵

∴

The equation of L2 is

∴

Quick Practice 5.9 (p. 5.21)

(a)

(b) (i) From (a), the coordinates of A and B are (10, 0) and

(0, 6) respectively.

Coordinates of the mid-point of AB

(ii) Let L1 be the perpendicular bisector of AB and m1 be

the slope of L1.

∵ L1⊥L

∴

∴ Slope of L1

∵ L1 is the perpendicular bisector of AB.

∴ L1 passes through the mid-point of AB.

The equation of L1 is

Quick Practice 5.10 (p. 5.27)

(a) Slope of L1

Slope of L2

∴ Slope of L1 = slope of L2 ……(1)

y-intercept of L1

y-intercept of L2

∴ y-intercept of L1 ≠ y-intercept of L2 ……(2)

By (1) and (2), L1 and L2 have no intersections.

(b) Slope of L1

Slope of L2

∴ Slope of L1 = slope of L2 ……(3)

y-intercept of L1

y-intercept of L2

∴ y-intercept of L1 = y-intercept of L2 ……(4)

By (3) and (4), L1 and L2 have an infinite number of intersections.

(c) Slope of L1

Slope of L2

∴ Slope of L1 ≠ slope of L2

∴ L1 and L2 have one intersection.

Quick Practice 5.11 (p. 5.28)

L1: 2x + y - 5 = 0 ……(1)

L2: x - y - 1 = 0 ……(2)

(1) + (2):

By substituting x = 2 into (2), we have

∴ Coordinates of P

Quick Practice 5.12 (p. 5.29)

(a) The equation of L2 is

∴

(b) L1: 2x + 4y + 15 = 0 ……(1)

L2: y = 2x ……(2)

(1) + (2):

By substituting y = -3 into (2), we have .

∴ Coordinates of P

(c) L3 passes through the point (0, 3) and .

∴

∴ The equation of L3 is y = 4x + 3.

Further Questions

Further Question (p. 5.19)

Slope of L2

Slope of L3

∵

∴ L3 is not perpendicular to L2.

Further Question (p. 5.20)

(a) Length of OM

(b) Coordinates of A and B are (8, 0) and (0, -6) respectively.

AB

(Pyth. theorem)

Consider the area of △AOB, we have

Further Question (p. 5.26)

∵ L1 and L2 do not intersect.

∴

Skills Consolidation

Skills Consolidation (p. 5.11)

1. (a) The equation of L is

(b) The equation of L is

2. Slope of L = –3

y-intercept = 4

The equation of L is .

3. (a) ∵ L passes through (1, 0) and (0, 1).

∴ The equation of L is

(b) By substituting (2, –1) into , we have

L.H.S. = -1

R.H.S. = -(2) + 1 = -1

∵ L.H.S. = R.H.S.

∴ (2, -1) satisfies the equation of L.

∴ L passes through A(2, -1).

Skills Consolidation (p. 5.21)

1. (a) (i)

∴ The equation of L1 is .

(ii)

(b) (i)

∴ The equation of L2 is .

(ii)

2. Let L be the required straight line and L1 be the straight line 5x + y + 6 = 0.

∵ L is parallel to the straight line 5x + y + 6 = 0.

∴

The equation of L is

3.  (a) Let (h, 0) and (0, k) be the coordinates of P and Q

respectively.

By substituting (h, 0) into 2x + 9y -18 = 0, we have

∴ Coordinates of P

By substituting (0, k) into 2x + 9y -18 = 0, we have

∴ Coordinates of Q

(b) Coordinates of the mid-point of PQ

The equation of the median from O to PQ in △is

∴

Skills Consolidation (p. 5.29)

1. (a) Slope of L1 = 1

Slope of L2 =

∴ Slope of L1 ≠ slope of L2

∴ L1 and L2 have one intersection.

(b) Slope of L1

Slope of L2

∴ Slope of L1 = slope of L2 ……(1)

y-intercept of L1

y-intercept of L2

∴ y-intercept of L1 ≠ y-intercept of L2 ……(2)

By (1) and (2), L1 and L2 have no intersections.

(c) Slope of L1

Slope of L2

∴ Slope of L1 = slope of L2 ……(3)

y-intercept of L1

y-intercept of L2

∴ y-intercept of L1 = y-intercept of L2 ……(4)

By (3) and (4), L1 and L2 have an infinite number of intersections.

2. L1: 3x - 5y + 11 = 0 ……(1)

L2: 2x + 3y - 18 = 0 ……(2)

2 ´ (1) - 3 ´ (2):

By substituting y = 4 into (2), we have

∴ Coordinates of P

Exercise

Exercise 5A (p. 5.13)

Level 1

1. The equation of the straight line L is

∴

2. The equation of the straight line L is

∴

3. ∵ y-intercept of the straight line L = -5

∴ The equation of the straight line L is.

4. ∵ y-intercept of the straight line L = 4

∴ The equation of the straight line L is.


5. The equation of the straight line L is

∴

6. ∵ y-intercept of the straight line L = -1

∴ The equation of the straight line L is

7. ∵ L is perpendicular to the x-axis.

∴ The equation of L is x = 4.

8. ∵ L is parallel to the x-axis.

∴ The equation of L is y = -1.

9. The equation of L is

∴

10. (a) The equation of the straight line is

∴

(b) The equation of the straight line is

∴

11. (a) The equation of the straight line is.

(b) The equation of the straight line is.

12. (a) ∵ y-intercept of the straight line = 5

∴ The equation of the straight line is.

(b) ∵ y-intercept of the straight line = -3

∴ The equation of the straight line is.

13. (a) The equation of the straight line is

∴


(b) The equation of the straight line is

∴

14. (a) The equation of the straight line is y = 4.

(b) The equation of the straight line is x = -3.

15. (a) The equation of the straight line is

∴

(b) The equation of the straight line is

∴

16. (a) The equation of the straight line is

∴

(b) The straight line is parallel to the y-axis.

∴ The equation of the straight line is .

17. (a) The straight line passes through the origin and (–2, 6).

The equation of the straight line is

∴

(b) The straight line passes through the origin and (4, 2). The equation of the straight line is

∴

18. (a)

∴ The slope is 1 and the y-intercept is –7.

(b)

∴ The slope is 2 and the y-intercept is –4.

(c)

∴ The slope is and the y-intercept is .

Level 2

19. Slope of L

The equation of L is

∴

20. Slope of L

y-intercept of L = 4

The equation of L is.

21. ∵ y-intercept of the straight line = 1

∴ The straight line passes through (0, 1).

The equation of the straight line is

22. ∵ The straight line passes through (2, 0) and (2, 2).

∴ The straight line is perpendicular to the x-axis.

∴ The equation of the straight line is x = 2.

23. ∵ x-intercept of the straight line = 3

∴ The straight line passes through (3, 0).

The equation of the straight line is

∴

24. The straight line passes through (-5, 0) and (0, 7).

The equation of the straight line is

25. Slope of the straight line

The equation of the straight line is

∴

26. (a) The equation of the straight line is

∴

(b) By substituting (–3, 17) into, we have

L.H.S. = 17

R.H.S.

∵ L.H.S. = R.H.S.

∴ The straight line passes through B(-3, 17).

27. (a) ∵ A(-4, 5) lies on the straight line L.

∴ By substituting (-4, 5) into, we have

(b) L:

∴ Slope of L

y-intercept of L

28. (a) L passes through (-4, 1) and .

The equation of L is

∴

(b) By substituting (k, 3) into, we

have

29. (a) ∵ The y-intercept and the slope of L are 3 and -4

respectively.

∴ The equation of L is .

(b) By substituting y = 0 into, we have

∴ The x-intercept of L is .

30. (a) (i) The coordinates of A are (-7, -2).

The coordinates of B are (-4, 6).

(ii) The equation of L is

∴

(b) The equation of L is .

∴ Slope of L

y-intercept of L

31. (a) The equation of L is

∴

(b) Coordinates of the mid-point of AB

The equation of L is

∴

32. (a) By substituting y = 0 into , we have