Introduction This Chapter focuses on coordinate geometry mainly

Introduction • This Chapter focuses on coordinate geometry, mainly involving straight line graphs • We will be looking at working out equations of graphs based on various sets of information

Coordinate Geometry in the (x, y) plane y Equation of a straight line The equation of a straight line is usually written in one of 2 forms. One you will have seen before; y-intercept gradient 1 x Where m is the gradient and c is the y-intercept. Or, the general form: Where a, b and c are integers. 5 A

Coordinate Geometry in the (x, y) plane Equation of a straight line The equation of a straight line is usually written in one of 2 forms. One you will have seen before; Example 1 Write down the gradient and yintercept of the following graphs a) Gradient = -3 y-intercept = (0, 2) Where m is the gradient and c is the y-intercept. b) Rearrange to get ‘y’ on one side Or, the general form: Where a, b and c are integers. Divide by 2 Gradient = 2 y-intercept = (0, 5/2) 5 A

Coordinate Geometry in the (x, y) plane Equation of a straight line The equation of a straight line is usually written in one of 2 forms. One you will have seen before; Where m is the gradient and c is the y-intercept. Or, the general form: Where a, b and c are integers. Example 2 Write each equation in the form ax + by + c = 0 a) b) -y Correct form +1/2 x and -5 x 2 (to remove fraction) Correct form 5 A

Coordinate Geometry in the (x, y) plane Equation of a straight line The equation of a straight line is usually written in one of 2 forms. One you will have seen before; Example 3 A line is parallel to the line y = 3 x + 2 and passes through (0, -1). Write the equation of the line. Parallel so the gradient will be the same Where m is the gradient and c is the y-intercept. Crosses through (0, -1), which is on the y-axis Or, the general form: 2 Where a, b and c are integers. -1 5 A

Coordinate Geometry in the (x, y) plane Equation of a straight line The equation of a straight line is usually written in one of 2 forms. One you will have seen before; Where m is the gradient and c is the y-intercept. Example 4 A line is parallel to the line 6 x + 3 y – 2 = 0 and passes through (0, 3). Write the equation of the line. Rearrange to the form y = mx + c Divide by 3 Or, the general form: Where a, b and c are integers. The new line has the same gradient, but intercepts the yaxis at 3 5 A

Coordinate Geometry in the (x, y) plane Equation of a straight line The equation of a straight line is usually written in one of 2 forms. One you will have seen before; Where m is the gradient and c is the y-intercept. Or, the general form: Where a, b and c are integers. Example 5 The line y = 4 x + 8 crosses the y-axis at 8. It crosses the x-axis at P. Work out the coordinates of P. Crosses the xaxis where y=0 -8 Divide by 4 So the line crosses the x-axis at (-2, 0) 5 A

Coordinate Geometry in the (x, y) plane The gradient of a line You can work out the gradient of a line if you know 2 points on it. Let the first point be (x 1, y 1) and the second be (x 2, y 2). The following formula gives the gradient: y (x 2, y 2) (x 1, y 1) y 2 - y 1 x 2 - x 1 x ‘The change in the y values, divided by the change in the x values’ 5 B

Coordinate Geometry in the (x, y) plane The gradient of a line You can work out the gradient of a line if you know 2 points on it. Let the first point be (x 1, y 1) and the second be (x 2, y 2). The following formula gives the gradient: Example 1 Calculate the gradient of the line which passes through (2, 3) and (5, 7) (x 1, y 1) = (2, 3) (x 2, y 2) = (5, 7) • ‘The change in the y values, divided by the change in the x values’ Substitute numbers in Work out or leave as a fraction 5 B

Coordinate Geometry in the (x, y) plane The gradient of a line You can work out the gradient of a line if you know 2 points on it. Let the first point be (x 1, y 1) and the second be (x 2, y 2). The following formula gives the gradient: ‘The change in the y values, divided by the change in the x values’ Example 2 Calculate the gradient of the line which passes through (-2, 7) and (4, 5) (x 1, y 1) = (-2, 7) (x 2, y 2) = (4, 5) Substitute numbers in Work out or leave as a fraction Simplify if possible 5 B

Coordinate Geometry in the (x, y) plane The gradient of a line You can work out the gradient of a line if you know 2 points on it. Let the first point be (x 1, y 1) and the second be (x 2, y 2). The following formula gives the gradient: ‘The change in the y values, divided by the change in the x values’ Example 3 Calculate the gradient of the line which passes through (2 d, -5 d) and (6 d, 3 d) (x 1, y 1) = (2 d, -5 d) (x 2, y 2) = (6 d, 3 d) Substitute numbers in Work out or leave as a fraction Simplify if possible (the d’s cancel out) 5 B

Coordinate Geometry in the (x, y) plane The gradient of a line You can work out the gradient of a line if you know 2 points on it. Let the first point be (x 1, y 1) and the second be (x 2, y 2). The following formula gives the gradient: Example 4 The line joining (2, -5) to (4, a) has a gradient of -1. Calculate the value of a. (x 1, y 1) = (2, -5) (x 2, y 2) = (4, a) Substitute numbers in Simplify ‘The change in the y values, divided by the change in the x values’ Multiply by 2 Subtract 5 5 B

Coordinate Geometry in the (x, y) plane Finding the Equation of a line You can find the equation of the line with gradient m, and coordinate (x 1, y 1) by using the following formula: Example 1 Find the equation of the line with gradient 5 that passes through the point (3, 2) (x 1, y 1) = (3, 2) m=5 Substitute the numbers in Expand the bracket Add 2 5 C

Coordinate Geometry in the (x, y) plane Finding the Equation of a line You can find the equation of the line with gradient m, and coordinate (x 1, y 1) by using the following formula: Example 2 Find the equation of the line with gradient 1/ that passes through the point (4, -6) 2 (x 1, y 1) = (4, -6) m = -1/2 Substitute the numbers in Expand the brackets Subtract 6 5 C

Coordinate Geometry in the (x, y) plane Finding the Equation of a line You can find the equation of the line with gradient m, and coordinate (x 1, y 1) by using the following formula: Example 3 The line y = 3 x – 9 crosses the x-axis at coordinate A. Find the equation of the line with gradient 2/3 that passes through A. Give your answer in the form ax + by + c = 0 where a, b and c are integers. At point A, y = 0 Thought Process ‘To find the equation of the line, I need point A’ ‘Point A is on the x-axis, so will have a y-coordinate of 0’ ‘As the equation I have already, crosses A as well, I can put y=0 into it to find out the x value at A’ Subtract 9 Divide by 3 A = (3, 0) 5 C

Coordinate Geometry in the (x, y) plane Finding the Equation of a line You can find the equation of the line with gradient m, and coordinate (x 1, y 1) by using the following formula: Example 3 A = (3, 0) The line y = 3 x – 9 crosses the x-axis at coordinate A. Find the equation of the line with gradient 2/3 that passes through A. Give your answer in the form ax + by + c = 0 where a, b and c are integers. (x 1, y 1) = (3, 0) m = 2 /3 Thought Process ‘To find the equation of the line, I need point A’ ‘Point A is on the x-axis, so will have a y-coordinate of 0’ ‘As the equation I have already, crosses A as well, I can put y=0 into it to find out the x value at A’ Substitute in values Multiply out bracket Subtract y Multiply by 3 5 C

Coordinate Geometry in the (x, y) plane Finding the Equation of a line You can find the equation of a line from 2 points by using the following formula: Example 1 Work out the equation of the line that goes through points (3, -1) and (5, 7). Give your answer in the form y = mx + c. (x 1, y 1) = (3, -1) (x 2, y 2) = (5, 7) Substitute in values Work out any sums Multiply the right side by 4 to make fractions the same Multiply by 8 Subtract 1 5 D

Coordinate Geometry in the (x, y) plane Finding the Equation of a line Example 1 You can find the equation of a line from 2 points by using the following formula: The lines y = 4 x – 7 and 2 x + 3 y – 21 = 0 intersect at point A. Point B has coordinates (-2, 8). Find the equation of the line that passes through A and B Thought Process ‘We need to find point A’ ‘If the equations intersect at A, they have the same value for y (and x)’ If I can write one of the equations in terms of y, I can replace the y in the second equation and solve it’ Replace y with ‘ 4 x - 7’ Expand the bracket Group x’s and add 42 Divide by 14 A = (3, 5) Sub x into one of the first equations to get y 5 D

Coordinate Geometry in the (x, y) plane Finding the Equation of a line Example 1 You can find the equation of a line from 2 points by using the following formula: The lines y = 4 x – 7 and 2 x + 3 y – 21 = 0 intersect at point A. Point B has coordinates (-2, 8). Find the equation of the line that passes through A and B (x 1, y 1) = (3, 5) A = (3, 5) (x 2, y 2) = (-2, 8) Substitute in values Thought Process ‘We need to find point A’ ‘If the equations intersect at A, they have the same value for y (and x)’ If I can write one of the equations in terms of y, I can replace the y in the second equation and solve it’ Work out the denominators Multiply all of left by -5 and all of right by 3 (makes denominators equal) Multiply by -15 Rearrange, keeping x positive 5 D

Coordinate Geometry in the (x, y) plane Finding the Perpendicular to a line You need to be able to work out the gradient of a line which is Perpendicular to another. Perpendicular means ‘intersects at a right angle… These lines are perpendicular Example 1 Work out the gradient of the line that is perpendicular to the lines with the following gradients. Line gradient Perpendicular 3 -1/3 1/ 2 -2 / 5 If a line has a gradient of m, the line perpendicular has gradient -1/m 2 x -2 5/ 2 -1/2 x Two perpendicular lines have gradients that multiply to give -1 5 E

Coordinate Geometry in the (x, y) plane Finding the Perpendicular to a line You need to be able to work out the gradient of a line which is Perpendicular to another. Perpendicular means ‘intersects at a right angle… These lines are perpendicular If a line has a gradient of m, the line perpendicular has gradient -1/m Two perpendicular lines have gradients that multiply to give -1 Example 2 Is the line y = 3 x + 4 perpendicular to the line x + 3 y – 3 = 0? Gradient = 3 Gradient = -1/3 The lines are perpendicular since their gradients multiply to give -1 5 E

Coordinate Geometry in the (x, y) plane Finding the Perpendicular to a line You need to be able to work out the gradient of a line which is Perpendicular to another. Perpendicular means ‘intersects at a right angle… These lines are perpendicular If a line has a gradient of m, the line perpendicular has gradient -1/m Two perpendicular lines have gradients that multiply to give -1 Example 3 Find an equation for the line that passes through (3, -1) and is perpendicular to the line y = 2 x - 4 Gradient = 2 Gradient of the perpendicular = -1/2 (x 1, y 1) = (3, -1) m = -1/2 Substitute in values Expand brackets Subtract 1 5 E

Summary • We have learnt how to write equations of a line in 2 different forms • We have done this from varying sets of information • We have also looked at the link between parallel and perpendicular lines