Coordinate Geometry I The Line Solutions Practice Questions

Coordinate Geometry I: The Line Solutions: Practice Questions 14. 8 CHAPTER 14

14 1. (i) Practice Questions 14. 8 Find the slope of each of the following lines: 2 x + y − 3 = 0 Method 1: Method 2: 2 x + y – 3 = 0 a = 2, b = 1 y = − 2 x + 3 Slope = − 2

14 1. (ii) Practice Questions 14. 8 Find the slope of each of the following lines: 5 x − y + 2 = 0 Method 1: Method 2: 5 x – y + 2 = 0 a = 5, b = − 1 – y = – 5 x – 2 y = 5 x + 2 Slope = 5

14 1. (iii) Practice Questions 14. 8 Find the slope of each of the following lines: 6 x + 2 y = 9 Method 1: Method 2: 6 x + 2 y = 9 a = 6, b = 2 2 y = − 6 x + 9

14 1. (iv) Practice Questions 14. 8 Find the slope of each of the following lines: 3 x − 7 y = − 4 Method 1: Method 2: 3 x – 7 y = − 4 a = 3, b = − 7 3 x + 4 = 7 y

14 1. (v) Practice Questions 14. 8 Find the slope of each of the following lines: x = 2 y + 3 Method 1: x = 2 y + 3 x – 2 y – 3 = 0 a = 1, b = − 2 Method 2: x = 2 y + 3 x – 3 = 2 y

14 1. (vi) Practice Questions 14. 8 Find the slope of each of the following lines: 2 y = 7 − 2 x Method 1: Method 2: 2 y = 7 – 2 x 0 = 2 x + 2 y – 7 a = 2, b = 2

14 2. (i) Practice Questions 14. 8 The equation of a line is 3 x − y + 5 = 0. Write this equation in the form y = mx + c and hence find: the slope of the line 3 x – y + 5 = 0 3 x + 5 = y y = 3 x + 5 Slope = 3

14 2. (ii) Practice Questions 14. 8 The equation of a line is 3 x − y + 5 = 0. Write this equation in the form y = mx + c and hence find: the point at which the line crosses the y-axis. When it crosses y-axis, x = 0 y = 3 x + 5 y = 3(0) + 5 y = 0 + 5 y = 5 (0, 5)

14 3. Practice Questions 14. 8 Which of the following lines are parallel to the line 6 x − 2 y + 5 = 0 6 x – 2 y + 5 = 0 (i) 3 y = 9 x − 2 a = 6, b = − 2 3 y = 9 x − 2 9 x – 3 y – 2 = 0 a = 9, b = − 3 Slope is equal to the slope of the given line, so the lines are parallel.

14 3. Practice Questions 14. 8 Which of the following lines are parallel to the line 6 x − 2 y + 5 = 0 6 x – 2 y + 5 = 0 (ii) 3 x − y − 3 = 0 a = 6, b = − 2 3 x – y – 3 = 0 a = 3, b = − 1 Slope is equal to the slope of the given line, so the lines are parallel.

14 3. Practice Questions 14. 8 Which of the following lines are parallel to the line 6 x − 2 y + 5 = 0 6 x – 2 y + 5 = 0 (iii) x + 3 y + 1 = 0 a = 1, b = 3 This line is perpendicular to the given line a = 6, b = − 2

14 3. Practice Questions 14. 8 Which of the following lines are parallel to the line 6 x − 2 y + 5 = 0 6 x – 2 y + 5 = 0 (iv) 4 y = 12 x − 7 a = 6, b = − 2 4 y = 12 x − 7 0 = 12 x – 4 y – 7 a = 12, b = − 4 Slope is equal to the slope of the given line, so the lines are parallel.

14 4. Practice Questions 14. 8 Show that the lines x + 2 y − 3 = 0 and 2 x + 4 y + 5 = 0 are parallel. x + 2 y – 3 = 0 2 x + 4 y + 5 = 0 a = 1, b = 2 a = 2, b = 4 The lines are parallel, since m 1 = m 2

14 5. Practice Questions 14. 8 Show that the lines 3 x − y + 7 = 0 and x + 3 y + 2 = 0 are perpendicular. 3 x – y + 7 = 0 x + 3 y + 2 = 0 a = 3, b = − 1 a = 1, b = 3 m 1 × m 2 Since the product of the slope is − 1, the lines are perpendicular.

14 6. Practice Questions 14. 8 If the line 5 x − 3 y + 7 = 0 is parallel to the line 10 x + ky − 1 = 0, find the value of k. 5 x − 3 y + 7 = 0 10 x + ky – 1 = 0 a = 5, b = − 3 a = 10, b = k Since the lines are parallel, their slopes are equal.

14 7. Practice Questions 14. 8 If the line x + 2 y − 2 = 0 is perpendicular to the line kx + y + 3 = 0, find the value of k. x + 2 y – 2 = 0 kx + y + 3 = 0 a = 1, b = 2 a = k, b = 1 For perpendicular lines, m 1 m 2 = − 1:

14 8. Practice Questions 14. 8 Find the equation of the line through (− 3, 1), which is parallel to the line 7 x + 2 y − 5 = 0. 7 x + 2 y – 5 = 0 a = 7, b = 2 Parallel lines have equal slopes m 1 = m 2 Therefore m 2 =

14 8. Practice Questions 14. 8 Find the equation of the line through (− 3, 1), which is parallel to the line 7 x + 2 y − 5 = 0. (x 1, y 1) = (− 3, 1) m = (y – y 1) = m(x – x 1) 7 x + 2 y + 19 = 0

14 9. Practice Questions 14. 8 Find the equation of the line through (3, 2), which is perpendicular to the line 4 x − y + 2 = 0. 4 x – y + 2 = 0 a = 4, b = – 1 Perpendicular lines, therefore:

14 9. Practice Questions 14. 8 Find the equation of the line through (3, 2), which is perpendicular to the line 4 x − y + 2 = 0. (x 1, y 1) = (3, 2) m = (y – y 1) = m(x – x 1) x + 4 y – 11 = 0

14 10. (i) Practice Questions 14. 8 l is the line 2 x − 5 y + 7 = 0. Find the equation of the line which is: parallel to the line l and passing through the point (2, − 1) Given line: 2 x – 5 y + 7 = 0 a = 2, b = − 5 Parallel lines have equal slopes, therefore m 1 = m 2

14 10. (i) Practice Questions 14. 8 l is the line 2 x − 5 y + 7 = 0. Find the equation of the line which is: parallel to the line l and passing through the point (2, − 1) (x 1, y 1) = (2, − 1) m 2 = 0 = 2 x – 5 y − 9

14 10. (ii) Practice Questions 14. 8 l is the line 2 x − 5 y + 7 = 0. Find the equation of the line which is: perpendicular to the line l and passing through the point (2, − 1). Perpendicular lines follow: m 1 m 2 = − 1 (x 1, y 1) = (2, − 1) m = 5 x + 2 y – 8 = 0

14 11. (i) Practice Questions 14. 8 n is the line 3 x + y − 8 = 0. Find the equation of the line which is: parallel to the line n and passing through the point (− 3, 1) Given line: 3 x + y – 8 = 0 a = 3, b = 1 Parallel lines have equal slopes, therefore m 1 = m 2 − 3 = m 2

14 11. (i) Practice Questions 14. 8 n is the line 3 x + y − 8 = 0. Find the equation of the line which is: parallel to the line n and passing through the point (− 3, 1) (x 1, y 1) = (− 3, 1) m = − 3 (y – y 1) = m(x – x 1) y – 1 = − 3(x – (− 3)) y – 1 = − 3(x + 3) y – 1 = − 3 x – 9 3 x + y + 8 = 0

14 11. (ii) Practice Questions 14. 8 n is the line 3 x + y − 8 = 0. Find the equation of the line which is: perpendicular to the line n and passing through the point (− 3, 1). Since the lines are perpendicular:

14 11. (ii) Practice Questions 14. 8 n is the line 3 x + y − 8 = 0. Find the equation of the line which is: perpendicular to the line n and passing through the point (− 3, 1). (x 1, y 1) = (− 3, 1) m = 0 = x – 3 y + 6