1 Edexcel New GCE A Level Maths workbook Straight line graphs Parallel and Perpendicular lines. Edited by: K V Kumaran kumarmaths.weebly.com2 Straight...

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Edited by: K V Kumaran

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Straight line graphs A LEVEL LINKS Scheme of work: 2a. Straight-line graphs, parallel/perpendicular, length and area problems

Key points • • •

A straight line has the equation y = mx + c, where m is the gradient and c is the y-intercept (where x = 0). The equation of a straight line can be written in the form ax + by + c = 0, where a, b and c are integers. When given the coordinates (x1, y1) and (x2, y2) of two points on a line the gradient is calculated using the y y1 formula m 2 x2 x1

Examples Example 1

A straight line has gradient

1 2

and y-intercept 3.

Write the equation of the line in the form ax + by + c = 0. m=

1 2

1 A straight line has equation y = mx + c. Substitute the gradient and y-intercept given in the question into this equation. 2 Rearrange the equation so all the terms are on one side and 0 is on the other side. 3 Multiply both sides by 2 to eliminate the denominator.

and c = 3 1 2

So y = x + 3 1 x 2

+y–3=0

x + 2y − 6 = 0

Example 2

Find the gradient and the y-intercept of the line with the equation 3y − 2x + 4 = 0. 3y − 2x + 4 = 0 3y = 2x − 4 2 4 y x 3 3 Gradient = m =

1 Make y the subject of the equation. 2 Divide all the terms by three to get the equation in the form y = …

2 3

y-intercept = c =

3 In the form y = mx + c, the gradient is m and the y-intercept is c.

4 3

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Example 3

Find the equation of the line which passes through the point (5, 13) and has gradient 3. m=3 y = 3x + c

1 Substitute the gradient given in the question into the equation of a straight line y = mx + c. 2 Substitute the coordinates x = 5 and y = 13 into the equation. 3 Simplify and solve the equation.

13 = 3 × 5 + c 13 = 15 + c c = −2 y = 3x − 2

4 Substitute c = −2 into the equation y = 3x + c

Example 4 Find the equation of the line passing through the points with coordinates (2, 4) and (8, 7). x1 2 , x2 8 , y1 4 and y2 7 y y1 7 4 3 1 m 2 x2 x1 8 2 6 2

1 xc 2 1 4 2c 2 c=3 1 y x3 2 y

1 Substitute the coordinates into the y y equation m 2 1 to work out x2 x1 the gradient of the line. 2 Substitute the gradient into the equation of a straight line y = mx + c. 3 Substitute the coordinates of either point into the equation. 4 Simplify and solve the equation. 5 Substitute c = 3 into the equation 1 y xc 2

Practice 1

2

Find the gradient and the y-intercept of the following equations. 1 2

a

y = 3x + 5

b

y= x–7

c

2y = 4x – 3

d

x + y = 5

e

2x – 3y – 7 = 0

f

5x + y – 4 = 0

Hint Rearrange the equations to the form y = mx + c

Copy and complete the table, giving the equation of the line in the form y = mx + c. Gradient

y-intercept

5

0

–3

2

4

–7

Equation of the line

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3

Find, in the form ax + by + c = 0 where a, b and c are integers, an equation for each of the lines with the following gradients and y-intercepts. 1 2

b

gradient 2, y-intercept 0

, y-intercept 4

d

gradient –1.2, y-intercept –2

a

gradient , y-intercept –7

c

gradient

2 3

4

Write an equation for the line which passes though the point (2, 5) and has gradient 4.

5

Write an equation for the line which passes through the point (6, 3) and has gradient

6

Write an equation for the line passing through each of the following pairs of points. a (4, 5), (10, 17) b (0, 6), (–4, 8) c (–1, –7), (5, 23) d (3, 10), (4, 7)

2 3

Extend 7

The equation of a line is 2y + 3x – 6 = 0. Write as much information as possible about this line.

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Answers 1

a

m = 3, c = 5

c

m = 2, c =

e

m=

2 3

3 2 7 3

, c = or –2

1 3

1 2

b

m = , c = –7

d

m = –1, c = 5

f

m = –5, c = 4

2

3

Gradient

y-intercept

Equation of the line

5

0

y = 5x

–3

2

y = –3x + 2

4

–7

y = 4x –7

a

x + 2y + 14 = 0

b

2x – y = 0

c

2x – 3y + 12 = 0

d

6x + 5y + 10 = 0

4

y = 4x – 3

5

y= x+7

6

a

y = 2x – 3

b

y= x+6

c

y = 5x –2

d

y = –3x + 19

2 3

1 2

3 3 y x 3 , the gradient is and the y-intercept is 3. 2 2 The line intercepts the axes at (0, 3) and (2, 0). 7

3 Students may sketch the line or give coordinates that lie on the line such as 1, or 4, 3 . 2

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Parallel and perpendicular lines Key points • •

When lines are parallel they have the same gradient. A line perpendicular to the line with equation 1 m

y = mx + c has gradient .

Examples Example 1

Find the equation of the line parallel to y = 2x + 4 which passes through the point (4, 9). y = 2x + 4 m=2 y = 2x + c 9=2×4+c 9=8+c c=1 y = 2x + 1

Example 2

1 As the lines are parallel they have the same gradient. 2 Substitute m = 2 into the equation of a straight line y = mx + c. 3 Substitute the coordinates into the equation y = 2x + c 4 Simplify and solve the equation. 5 Substitute c = 1 into the equation y = 2x + c

Find the equation of the line perpendicular to y = 2x − 3 which passes through the point (−2, 5). y = 2x − 3 m=2 1 1 m 2 1 y xc 2 1 5 ( 2) c 2 5=1+c c=4 1 y x4 2

1 As the lines are perpendicular, the gradient of the perpendicular line 1 is . m 1 2 Substitute m = into y = mx + c. 2 3 Substitute the coordinates (–2, 5) 1 into the equation y x c 2 4 Simplify and solve the equation. 1 5 Substitute c = 4 into y x c . 2

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Example 3

A line passes through the points (0, 5) and (9, −1). Find the equation of the line which is perpendicular to the line and passes through its midpoint. x1 0 , x2 9 , y1 5 and y2 1

m

y2 y1 1 5 x2 x1 90

1 Substitute the coordinates into the y y1 equation m 2 to work out x2 x1 the gradient of the line.

6 2 9 3

2 As the lines are perpendicular, the gradient of the perpendicular line 1 is . m 3 Substitute the gradient into the equation y = mx + c.

1 3 m 2

y

3 xc 2

0 9 5 ( 1) 9 , Midpoint = , 2 2 2 2 3 9 2 c 2 2 19 c 4 3 19 y x 2 4

4 Work out the coordinates of the midpoint of the line. 5 Substitute the coordinates of the midpoint into the equation. 6 Simplify and solve the equation. 19 7 Substitute c into the equation 4 3 y xc. 2

Practice 1

Find the equation of the line parallel to each of the given lines and which passes through each of the given points. a y = 3x + 1 (3, 2) b y = 3 – 2x (1, 3) c 2x + 4y + 3 = 0 (6, –3) d 2y –3x + 2 = 0 (8, 20)

2

Find the equation of the line perpendicular to y =

1 2

x – 3 which

passes through the point (–5, 3).

Hint If m =

then the

negative reciprocal

3

a b

1 b m a

Find the equation of the line perpendicular to each of the given lines and which passes through each of the given points. 1 3

1 2

a

y = 2x – 6 (4, 0)

b

y = x +

(2, 13)

c

x –4y – 4 = 0 (5, 15)

d

5y + 2x – 5 = 0 (6, 7) kumarmaths.weebly.com

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4

In each case find an equation for the line passing through the origin which is also perpendicular to the line joining the two points given. a (4, 3), (–2, –9) b (0, 3), (–10, 8)

Extend 5

Work out whether these pairs of lines are parallel, perpendicular or neither. a y = 2x + 3 b y = 3x c y = 4x – 3 y = 2x – 7 2x + y – 3 = 0 4y + x = 2 d

6

3x – y + 5 = 0 x + 3y = 1

e

2x + 5y – 1 = 0 y = 2x + 7

f

2x – y = 6 6x – 3y + 3 = 0

The straight line L1 passes through the points A and B with coordinates (–4, 4) and (2, 1), respectively. a Find the equation of L1 in the form ax + by + c = 0 The line L2 is parallel to the line L1 and passes through the point C with coordinates (–8, 3). b Find the equation of L2 in the form ax + by + c = 0 The line L3 is perpendicular to the line L1 and passes through the origin. c Find an equation of L3

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Answers 1

a

y = 3x –7

b

y = –2x + 5

c

y = –1 x

d

y=

2

3 2

x+8

2

y = −2x – 7

3

a

y = –1 x + 2

b

y = 3x + 7

c

y = –4x + 35

d

y=

4

a

y = – 1 x

b

y = 2x

5

a d

Parallel Perpendicular

b e

Neither Neither

c f

Perpendicular Parallel

6

a

x + 2y – 4 = 0

b

x + 2y + 2 = 0

c

y = 2x

2

2

5 2

x–8

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Q1. The point A (–6, 4) and the point B (8, –3) lie on the line L. (a) Find an equation for L in the form ax + by + c = 0, where a, b and c are integers. (4) (b) Find the distance AB, giving your answer in the form k√5, where k is an integer. (3)

Q2. The points P and Q have coordinates (−1, 6) and (9, 0) respectively. The line l is perpendicular to PQ and passes through the mid-point of PQ. Find an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are integers. (5)

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Q3. The line l1 has equation 3x + 5y − 2 = 0 (a) Find the gradient of l1. (2) The line l2 is perpendicular to l1 and passes through the point (3, 1). (b) Find the equation of l2 in the form y = mx + c, where m and c are constants. (3)

Q4. The line l1 has equation y = − 2x + 3 The line l2 is perpendicular to l1 and passes through the point (5, 6). (a) Find an equation for l2 in the form ax + by + c = 0, where a , b and c are integers. (3) The line l2 crosses the x-axis at the point A and the y-axis at the point B. (b) Find the x-coordinate of A and the y-coordinate of B. (2)

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Q5.

Figure 1 The line l1 has equation 2x − 3y + 12 = 0 (a) find the gradient of l1. (1) The line l1 crosses the x-axis at the point A and the y-axis at the point B, as shown in Figure 1. The line l2 is perpendicular to l1 and passes through B. (b) Find an equation of l2. (3) The line l2 crosses the x-axis at the point C. (c) Find the area of triangle ABC. (4)

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Q6. The line L1 has equation 2y − 3x − k = 0, where k is a constant. Given that the point A (1, 4) lies on L1, find (a) the value of k, (1) (b) the gradient of L1. (2) The line L2 passes through A and is perpendicular to L1 (c) Find an equation of L2 giving your answer in the form ax + by + c = 0, where a, b and c are integers. (4) The line L2 crosses the x-axis at the point B. (d) Find the coordinates of B. (2) (e) Find the exact length of AB. (2)

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Q7.

The points A and B have coordinates (6, 7) and (8, 2) respectively. The line l passes through the point A and is perpendicular to the line AB, as shown in Figure 1. (a) Find an equation for l in the form ax + by + c = 0, where a, b and c are integers. (4) Given that l intersects the y-axis at the point C, find (b) the coordinates of C, (2) (c) the area of ΔOCB, where O is the origin. (2)

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Q8.

Figure 2 The line l1, shown in Figure 2 has equation 2x + 3y = 26 The line l2 passes through the origin O and is perpendicular to l1 (a) Find an equation for the line l2 (4) The line l2 intersects the line l1 at the point C. Line l1 crosses the y-axis at the point B as shown in Figure 2. (b) Find the area of triangle OBC. Give your answer in the form a⁄b, where a and b are integers to be determined. (6)

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Q9.

Figure 2 Figure 2 shows a right angled triangle LMN. The points L and M have coordinates (−1, 2) and (7, −4) respectively. (a) Find an equation for the straight line passing through the points L and M. Give your answer in the form ax + by + c = 0, where a, b and c are integers. (4) Given that the coordinates of point N are (16, p), where p is a constant, and angle LMN = 90°, (b) find the value of p. (3) Given that there is a point K such that the points L, M, N, and K form a rectangle, (c) find the y coordinate of K. (2)

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Q10.

The points P (0, 2) and Q (3, 7) lie on the line l1, as shown in Figure 2. The line l2 is perpendicular to l1, passes through Q and crosses the x-axis at the point R, as shown in Figure 2. Find (a) an equation for l2, giving your answer in the form ax + by + c = 0, where a, b and c are integers, (5) (b) the exact coordinates of R, (2) (c) the exact area of the quadrilateral ORQP, where O is the origin. (5)

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