COORDINATE GEOMETRY Straight Lines The equations of straight lines come in two forms: 1.y = mx + c,...

COORDINATE GEOMETRY

Straight Lines

The equations of straight lines come in two forms:

1. y = mx + c, where m is the gradient and c is the y-intercept.

2. ax + by + c = 0, where a, c and c are integers.

COORDINATE GEOMETRY

Straight Lines

When equations are in the form ax + by + c = 0 they can be rearranged into the form y = mx + c so that the gradient and the y-intercept can be found easily.

Example:Write 2x + 3y + 5 = 0 in the form y = mx + c

and state the gradient and y-intercept of the line.

COORDINATE GEOMETRY

Straight Lines

Points of Revision:

To find where a line cuts the x-axis let the equation equal 0.

To find where a line cuts the y-axis let x equal 0 in the equation.

When two lines are parallel they have the same gradient.

COORDINATE GEOMETRY

Straight Lines

Examples:Write these lines in the form ax + by + c = 0

(i) y = 2x + 3 (ii) y = 1/4x – 3

COORDINATE GEOMETRY

Straight Lines

Examples:A line is parallel to the line y = 1/3x – 4 and

crosses the y-axis at the point (0, 6). Write down the equation of the line.

COORDINATE GEOMETRY

Straight Lines

Examples:A line is parallel to the line 3x + 5y + 1 = 0 and

it passes through the point (0, 4). Work out the equation of the line.

COORDINATE GEOMETRY

Straight Lines

Examples:The line y = 3x – 12 meets the x-axis at the

point P. Find the coordinates of P.

COORDINATE GEOMETRY

Straight Lines

Finding the gradient when given two points

If given two points on a line, (x1, y1) and (x2, y2), we can find the gradient of the line by using

the formula:

y2 – y1

m = -------- x2 – x1

COORDINATE GEOMETRY

Straight Lines

Examples:Work out the gradient of the line joining the

following points:

(i)(3, 4) and (5, 6) (ii) (3a, -2a) and (4a, 2a)

COORDINATE GEOMETRY

Straight Lines

Examples:The line joining (2, -5) to (4, a) has gradient -1. Work out the value of a.

COORDINATE GEOMETRY

Finding the Equation of a Straight Line

If given the gradient of a line and a point, (x1, y1), on the line we can find the equation of line using the formula:

y – y1 = m(x – x1)

COORDINATE GEOMETRY

Straight Lines

Examples:Find the equation of the line with gradient 4 that passes through the point (1, 3).

COORDINATE GEOMETRY

Straight Lines

Examples:Find the equation of the line with gradient -½ that passes through the point (5, 3).

COORDINATE GEOMETRY

Straight Lines

Examples:The line y = 4x – 8 meets the x-axis at the point A. Find the equation of the line with gradient 3 that passes through the point A.

COORDINATE GEOMETRY

Finding the Equation of a Straight Line

If given the two points on a line, (x1, y1) and (x2, y2), we can find the equation of line using the formula:

y – y1 x – x1

------- = ------- y2 – y1 x2 – x1

COORDINATE GEOMETRY

Straight Lines

Examples:The find the equation of the line that passes through the points (1, 2) and (5, 4).

COORDINATE GEOMETRY

Straight Lines

Examples:The lines y = 4x – 7 and 2x + 3y -21 = 0 intersect at the point A. The point B has coordinates (-2, 8). Find the equation of the line that passes through the points A and B. Write your answer in the form ax + by + c = 0.

COORDINATE GEOMETRY

Perpendicular Lines

If two lines are perpendicular then

Gradient of line 1 x Gradient of line 2 = -1

Thus, if the gradient of line 1 = m, then the gradient of line 2 = -1/m

COORDINATE GEOMETRY

Examples:Work out the gradient of the line that is perpendicular to the lines with these gradients:

(i) 3 (ii) -4 (iii) -½

COORDINATE GEOMETRY

Examples:Show that the lines y = 2x + 5 and x + 2y + 6 = 0 are perpendicular.

COORDINATE GEOMETRY

Examples:Determine whether the lines y – 3x + 3 = 0 and 3y + x = 6 are parallel, perpendicular or neither.

COORDINATE GEOMETRY

Examples:Line L is perpendicular to the line 2y – x + 3 = 0 at the point (4, ½). Determine the equation of the line L.

COORDINATE GEOMETRY

SUMMARYEquations of lines can be written in the form:ax + by + c = 0 or y = mx + c

Given two points we can find the gradient of the line joining the points using the formula

y2 – y1

m = -------- x2 – x1

COORDINATE GEOMETRY

SUMMARY

Given a point on a line and its gradient we can find the equation of the line using the formula:

y – y1 = m(x – x1)

Given two points on a line we can find the equation of the line using the formula:

y – y1 x – x1

------- = ------- y2 – y1 x2 – x1