Edexcel New GCE A Level Maths workbook

Size: px

Start display at page:

Download "Edexcel New GCE A Level Maths workbook"

Jack Powell
6 years ago
Views:

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

2 Straight line graphs A LEVEL LINKS Scheme of work: a. 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 (x, y ) and (x, y ) of two points on a line the gradient is calculated using the y y formula m x x Examples Example A straight line has gradient and y-intercept. Write the equation of the line in the form ax + by + c = 0. m = and c = So y = x + x + y = 0 x + y 6 = 0 A straight line has equation y = mx + c. Substitute the gradient and y-intercept given in the question into this equation. Rearrange the equation so all the terms are on one side and 0 is on the other side. Multiply both sides by to eliminate the denominator. Example Find the gradient and the y-intercept of the line with the equation y x + 4 = 0. y x + 4 = 0 y = x 4 4 y x Gradient = m = 4 y-intercept = c = Make y the subject of the equation. Divide all the terms by three to get the equation in the form y = In the form y = mx + c, the gradient is m and the y-intercept is c. kumarmaths.weebly.com

3 Example Find the equation of the line which passes through the point (5, ) and has gradient. m = y = x + c = 5 + c = 5 + c c = y = x Substitute the gradient given in the question into the equation of a straight line y = mx + c. Substitute the coordinates x = 5 and y = into the equation. Simplify and solve the equation. 4 Substitute c = into the equation y = x + c Example 4 Find the equation of the line passing through the points with coordinates (, 4) and (8, 7). x, x 8, y 4 and y 7 y y 7 4 m x x 8 6 y x c 4 c c = y x Substitute the coordinates into the y y equation m to work out x x the gradient of the line. Substitute the gradient into the equation of a straight line y = mx + c. Substitute the coordinates of either point into the equation. 4 Simplify and solve the equation. 5 Substitute c = into the equation y x c Practice Find the gradient and the y-intercept of the following equations. a y = x + 5 b y = x 7 c y = 4x d x + y = 5 e x y 7 = 0 f 5x + y 4 = 0 Copy and complete the table, giving the equation of the line in the form y = mx + c. Gradient y-intercept Equation of the line Hint Rearrange the equations to the form y = mx + c kumarmaths.weebly.com

4 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. a gradient, y-intercept 7 b gradient, y-intercept 0 c gradient, y-intercept 4 d gradient., y-intercept 4 Write an equation for the line which passes though the point (, 5) and has gradient 4. 5 Write an equation for the line which passes through the point (6, ) and has gradient 6 Write an equation for the line passing through each of the following pairs of points. a (4, 5), (0, 7) b (0, 6), ( 4, 8) c (, 7), (5, ) d (, 0), (4, 7) Extend 7 The equation of a line is y + x 6 = 0. Write as much information as possible about this line. kumarmaths.weebly.com 4

5 Answers a m =, c = 5 b m = c m =, c = e m =, c = 7 or, c = 7 d m =, c = 5 f m = 5, c = 4 Gradient y-intercept Equation of the line 5 0 y = 5x y = x y = 4x 7 a x + y + 4 = 0 b x y = 0 c x y + = 0 d 6x + 5y + 0 = 0 4 y = 4x 5 y = x a y = x b y = x + 6 c y = 5x d y = x y x, the gradient is and the y-intercept is. The line intercepts the axes at (0, ) and (, 0). Students may sketch the line or give coordinates that lie on the line such as, or 4,. kumarmaths.weebly.com 5

6 Parallel and perpendicular lines Key points When lines are parallel they have the same gradient. A line perpendicular to the line with equation y = mx + c has gradient. m Examples Example Find the equation of the line parallel to y = x + 4 which passes through the point (4, 9). y = x + 4 m = y = x + c 9 = 4 + c 9 = 8 + c c = y = x + As the lines are parallel they have the same gradient. Substitute m = into the equation of a straight line y = mx + c. Substitute the coordinates into the equation y = x + c 4 Simplify and solve the equation. 5 Substitute c = into the equation y = x + c Example Find the equation of the line perpendicular to y = x which passes through the point (, 5). y = x m = m y x c 5 ( ) 5 = + c c = 4 y x 4 c As the lines are perpendicular, the gradient of the perpendicular line is. m Substitute m = into y = mx + c. Substitute the coordinates (, 5) into the equation y x c 4 Simplify and solve the equation. 5 Substitute c = 4 into y x c. kumarmaths.weebly.com 6

7 Example A line passes through the points (0, 5) and (9, ). Find the equation of the line which is perpendicular to the line and passes through its midpoint. x 0, x 9, y 5 and y y y 5 m x x 9 0 m y x c ( ) 9 Midpoint =,, 9 c 9 c 4 9 y x 4 Substitute the coordinates into the y y equation m to work out x x the gradient of the line. As the lines are perpendicular, the gradient of the perpendicular line is. m Substitute the gradient into the equation y = mx + c. 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. 9 7 Substitute c into the equation 4 y x c. Practice Find the equation of the line parallel to each of the given lines and which passes through each of the given points. a y = x + (, ) b y = x (, ) c x + 4y + = 0 (6, ) d y x + = 0 (8, 0) Find the equation of the line perpendicular to y = x which passes through the point ( 5, ). Hint If m = a then the b negative reciprocal 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. a y = x 6 (4, 0) b y = x + (, ) c x 4y 4 = 0 (5, 5) d 5y + x 5 = 0 (6, 7) kumarmaths.weebly.com 7

8 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, ), (, 9) b (0, ), ( 0, 8) Extend 5 Work out whether these pairs of lines are parallel, perpendicular or neither. a y = x + b y = x c y = 4x y = x 7 x + y = 0 4y + x = d x y + 5 = 0 e x + 5y = 0 f x y = 6 x + y = y = x + 7 6x y + = 0 6 The straight line L passes through the points A and B with coordinates ( 4, 4) and (, ), respectively. a Find the equation of L in the form ax + by + c = 0 The line L is parallel to the line L and passes through the point C with coordinates ( 8, ). b Find the equation of L in the form ax + by + c = 0 The line L is perpendicular to the line L and passes through the origin. c Find an equation of L kumarmaths.weebly.com 8

9 Answers a y = x 7 b y = x + 5 c y = x d y = x + 8 y = x 7 a y = x + b y = x + 7 c y = 4x + 5 d y = 5 x 8 4 a y = x b y = x 5 a Parallel b Neither c Perpendicular d Perpendicular e Neither f Parallel 6 a x + y 4 = 0 b x + y + = 0 c y = x kumarmaths.weebly.com 9

10 Q. The point A ( 6, 4) and the point B (8, ) 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. (b) Find the distance AB, giving your answer in the form k 5, where k is an integer. (4) () Q. The points P and Q have coordinates (, 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) kumarmaths.weebly.com 0

11 Q. The line l has equation x + 5y = 0 (a) Find the gradient of l. The line l is perpendicular to l and passes through the point (, ). (b) Find the equation of l in the form y = mx + c, where m and c are constants. () () Q4. The line l has equation y = x + The line l is perpendicular to l and passes through the point (5, 6). (a) Find an equation for l in the form ax + by + c = 0, where a, b and c are integers. The line l 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. () () kumarmaths.weebly.com

12 Q5. Figure The line l has equation x y + = 0 (a) find the gradient of l. () The line l crosses the x-axis at the point A and the y-axis at the point B, as shown in Figure. The line l is perpendicular to l and passes through B. (b) Find an equation of l. () The line l crosses the x-axis at the point C. (c) Find the area of triangle ABC. (4) kumarmaths.weebly.com

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

14 Q7. The points A and B have coordinates (6, 7) and (8, ) respectively. The line l passes through the point A and is perpendicular to the line AB, as shown in Figure. (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, () (c) the area of ΔOCB, where O is the origin. () kumarmaths.weebly.com 4

15 Q8. Figure The line l, shown in Figure has equation x + y = 6 The line l passes through the origin O and is perpendicular to l (a) Find an equation for the line l The line l intersects the line l at the point C. Line l crosses the y-axis at the point B as shown in Figure. (b) Find the area of triangle OBC. Give your answer in the form a b, where a and b are integers to be determined. (4) (6) kumarmaths.weebly.com 5

16 Q9. Figure Figure shows a right angled triangle LMN. The points L and M have coordinates (, ) 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 (6, p), where p is a constant, and angle LMN = 90, (b) find the value of p. () 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. () kumarmaths.weebly.com 6

17 Q0. The points P (0, ) and Q (, 7) lie on the line l, as shown in Figure. The line l is perpendicular to l, passes through Q and crosses the x-axis at the point R, as shown in Figure. Find (a) an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are integers, (b) the exact coordinates of R, (c) the exact area of the quadrilateral ORQP, where O is the origin. (5) () (5) kumarmaths.weebly.com 7

NAME: Date: HOMEWORK: C1. Question Obtained. Total/100 A 80 B 70 C 60 D 50 E 40 U 39

NAME: Date: HOMEWORK: C1. Question Obtained. Total/100 A 80 B 70 C 60 D 50 E 40 U 39 NAME: Date: HOMEWORK: C1 Question Obtained 1 2 3 4 5 6 7 8 9 10 Total/100 A 80 B 70 C 60 D 50 E 40 U 39 1. Figure 2 y A(1, 7) B(20, 7) D(8, 2) O x C(p, q) The points A(1, 7), B(20, 7) and C(p, q) form