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

The equation of a straight line can be written as y = mx + c, where m is the gradient and c is the intercept on the y-axis. It can also be found from a gradient and one point, or from two points. The gradients of two lines show whether they are parallel or perpendicular.

Behavioural objectives

By the end of the lesson, students should be able to:

• state the gradient and y-intercept of a line from its equation;
• find the equation of a line given its gradient and one point;
• find the equation of a line given two points;
• show that two lines are parallel or perpendicular using their gradients;
• calculate the acute angle between two intersecting lines.

What are the gradient and intercept of a line?

In a line of the form y = mx + c, the gradient is the coefficient of x, represented by m. The letter 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 are the gradient and y-intercept of the line 3x – 5y + 10 = 0?
• Find the equation of the line with gradient -9 and y-intercept 4.

How do you find the equation of a line from its gradient and one point?

The equation of a line can be calculated from one point (x₁, y₁) and the 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)
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).

How do you find the equation of a line from two points?

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

(y₂ – y₁)/(x₂ – x₁) = (y – y₁)/(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)
11/1 = (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).

What is the angle between two lines?

Parallel lines

The angle between parallel lines is 0° because they have the same gradient.

Perpendicular lines

The angle between two perpendicular lines is 90°, and the product of their gradients is -1. Hence, m₁m₂ = -1.

Example 1

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

Solution

Equation 1: y = -3x + 2, so m₁ = -3
Equation 2: y + 3x = 7
y = -3x + 7, so m₂ = -3

Since m₁ = m₂ = -3, the lines are parallel.

Example 2

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

Solution

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, so m₂ = -3

Hence, m₁ × m₂ = 1/3 × (-3) = -1.

Since m₁m₂ = -1, the lines are perpendicular.

Evaluation

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

Angle between intersecting lines

The gradient of y = mx + c is tan θ. Hence, m = tan θ can be used to calculate the angle between two intersecting lines. In general, the angle between two lines is obtained using:

tan θ = |(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, so m₁ = 4
y = x/2 + 0.5, so m₂ = 1/2 = 0.5

tan θ = (0.5 – 4)/(1 + (0.5 × 4)) = -3.5/3 = -1.1667
Taking the positive value for the acute angle, tan θ = 1.1667
θ = tan^-1(1.1667) = 49.4°

Evaluation

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

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) with gradient 3.

Assignment

Reading assignment: NGM for SS 3, Chapter 9, pages 75 to 81.

Weekend assignment

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