Coordinate Geometry of a Straight Line – SS3 Mathematics Lesson Note (Second Term)

Gradient and Intercepts of a line

A gradient of a line of the form y = mx + c is the coefficient of x, which is represented by m and c is the intercept on the y-axis.

Example

• Find the equation of the line with gradient 4 and y-intercept -7.

Solution

m = 4, c = – 7,

Hence, the equation is; y =4x – 7.

Evaluation: 

• What is the gradient and y-intercept of the line equation 3x -5y +10=0?
• Find the equation of the line with gradient – 9 and y-intercept 4.

Gradient and One-Point Form

The equation of the line can be calculated given one point (x, y) and gradient (m) by using the formula; y – y1= m(x – x1)

Example

Find the equation of the line with gradient -8 and point(3, 7).

Solution

m = – 8, (x1, y1) =(3,7)

Equation: y – 7 = – 8(x – 3)

                 y = -8x + 24 +7

                 y = -8x + 31 

Evaluation: 

• Find the equation of the line with gradient 5 and point(-2, -7).
• Find the equation of the line with gradient -12 and point (3, -5).

 Two Point Form:

Given two points (x1, y1) and (x2, y2), the equation can be obtained using the formula:

 y2 – y1 = y – y1

x2 – x1      x – x1

Example: Find the equation of the line passing through (2,-5) and (3,6).

Solution

6 – (-5)/3 – 2 = y – (-5)/x – 2

11 = y + 5/x – 2

11(x – 2) = y + 5

11x – 22 = y + 5

y – 11x + 27 = 0

Evaluation: 

• Find the equation of the line passing through (3, 4) and (-1, -2).
• Find the equation of the line passing through (-8, 5) and (-6, 2).

 Angles between Lines

Parallel lines:

The angle between parallel lines is 00 because they have the same gradient

Perpendicular Lines:

The angle between two perpendicular lines is 900 and the product of their gradients is – 

1. Hence, m1m2 = – 1

Examples: 

1. Show that the lines y = -3x + 2 and y + 3x = 7 are parallel.

Solution:

         Equation 1: y = -3x + 2,   m1 = -3

         Equation 2:  y + 3x = 7,   

                                 y = -3x + 7, m2 = – 3

since; m1 = m2 = – 3, then the lines are parallel

1. Given the line equations x = 3y + 5 and y + 3x = 2, show that the lines are perpendicular.

Solutions:

     Equation 1:     x = 3y + 5,   make y the subject of the equation.

                              3y = x + 5

                                y = x/3 + 5/3

                            m1 = 1/3 

  Equation 2:    y + 3x = 2,

                              y = – 3x + 2,   m2 = -3 

hence: m1 x m2 = 1/3 x – 3 = – 1 

since: m1m2 = – 1, then the lines are perpendicular.

 Evaluation: State which of the following pairs of lines are: (i) perpendicular   (ii) parallel

Angles between Intersecting Lines:

The gradient of y = mx + c is tan θ.    Hence m = tan θcan be used to calculate angles between two intersecting lines. Generally, the angle between two lines can be obtained using: tan 0 = m2 -m1

1 + m1m2

Example: Calculate the acute angle between the lines y=4x -7 and y = x/2 + 0.5.

Solution: 

Y=4x -7, m1= 4, y=x/2+0.2, m2 =1/2.

 Tan O= 0.5 – 4.       = -3.5/3

                    1 + (0.5*4)

Tan O =- 1.1667 

O=tan-1(-1.1667) = 49.4

Evaluation: Calculate the acute angle between the lines y=3x -4 and x – 4y +8 = 0.

General Evaluation:

• Calculate the acute angle between the lines y=2x -1 and  2y + x = 2.
• If the lines 3y=4x -1 and qy= x + 3 are parallel to each other, find the value of q.
• Find the equation of the line passing through (2,-1) and gradient 3.

Reading Assignment: NGM for SS 3 Chapter 9 pages 75 – 81 

Weekend Assignment

• Find the equation of the line passing through (5,0) and gradient 3.
• Find the equation of the line passing through (2,-1) and (1, -2).
• Two lines y=3x – 4 and x – 4y + 8=0 are drawn on the same axes.
• Find the gradients and intercepts on the axes of each line.
• Find the equation parallel to x -4y + 8=0 at the point (3, -5)