General form of a line; parallel and perpendicular lines
Every line can be written in the general form ax + by + c = 0. Its slope is −a/b, and parallel or perpendicular lines follow from it.
Reading the line 2x + 3y − 6 = 0
| What you want | Rule | For 2x + 3y − 6 = 0 |
|---|---|---|
| Slope | −a/b | −2/3 |
| x-intercept | −c/a (put y = 0) | −(−6)/2 = 3 |
| y-intercept | −c/b (put x = 0) | −(−6)/3 = 2 |
| A parallel line | Keep a and b, change c to k | 2x + 3y + k = 0 |
| A perpendicular line | Swap a and b, change one sign | 3x − 2y + k = 0 |
Two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Your textbook shows these by comparing the slopes −a/b, so write the slopes in your answer. The two tests above are a quick check. To find k, put the given point into ax + by + k = 0 (or bx − ay + k = 0) and solve for k.
Example 1: slope and intercepts
Find the slope, x-intercept and y-intercept of the line 3x − 4y + 12 = 0.
- Here a = 3, b = −4 and c = 12
- Slope = −a/b = −3/(−4) = 3/4
- x-intercept = −c/a = −12/3 = −4
- y-intercept = −c/b = −12/(−4) = 3
- Check: (−4, 0) gives −12 + 12 = 0, and (0, 3) gives −12 + 12 = 0
Slope = 3/4, x-intercept = −4, y-intercept = 3
Example 2: parallel and perpendicular lines through a point
Find the equations of the lines through P(3, −1) that are (i) parallel and (ii) perpendicular to 2x − 5y + 4 = 0.
- (i) A parallel line has the form 2x − 5y + k = 0
- It passes through P: 2(3) − 5(−1) + k = 0, so 6 + 5 + k = 0 and k = −11
- Parallel line: 2x − 5y − 11 = 0
- (ii) Swap 2 and −5 and change one sign: a perpendicular line has the form 5x + 2y + k = 0
- Through P: 5(3) + 2(−1) + k = 0, so 15 − 2 + k = 0 and k = −13
- Perpendicular line: 5x + 2y − 13 = 0
(i) 2x − 5y − 11 = 0 (ii) 5x + 2y − 13 = 0
Example 3: will the road and the canal meet?
On a map, a road runs along 3x + 2y − 6 = 0 and a canal along 6x + 4y + 5 = 0. Will they ever meet?
- Formula: the slope of ax + by + c = 0 is −a/b
- Slope of the road = −3/2. Slope of the canal = −6/4 = −3/2
- The slopes are equal, so the lines are parallel.
- Quick check: a₁b₂ − a₂b₁ = 3 × 4 − 6 × 2 = 12 − 12 = 0
- They are not the same line: 2 × (3x + 2y − 6) = 6x + 4y − 12, which is not 6x + 4y + 5.
The road and the canal are parallel, so they never meet.
Engineers keep a new road next to a railway line parallel to it, so the slopes match. A road at a level crossing is usually laid perpendicular to the track.
2 marks: “Show that the lines 3x − 4y + 7 = 0 and 6x − 8y − 5 = 0 are parallel.” Slope of the first line = −3/(−4) = 3/4. Slope of the second = −6/(−8) = 3/4. The slopes are equal, so the lines are **parallel**. (Quick check: a₁b₂ − a₂b₁ = 3(−8) − 6(−4) = 0.)
2 marks: “Find k if the lines kx + 3y − 2 = 0 and 2x − y + 3 = 0 are perpendicular.” Slopes: −k/3 and −2/(−1) = 2. Perpendicular, so (−k/3) × 2 = −1, giving **k = 3/2**. (Quick check: a₁a₂ + b₁b₂ = 2k + 3(−1) = 0.)
5 marks: “Find the equations of the lines through P(−1, 4) that are (i) parallel and (ii) perpendicular to 3x − 2y + 6 = 0.” (i) 3x − 2y + k = 0 through P: −3 − 8 + k = 0, so k = 11 and the line is **3x − 2y + 11 = 0**. (ii) 2x + 3y + k = 0 through P: −2 + 12 + k = 0, so k = −10 and the line is **2x + 3y − 10 = 0**.
Parallel: keep a and b. Perpendicular: swap a and b and flip one sign. Then find k from the point.