- Two lines are parallel if and only if they have the same slope. They never intersect.
- If m1 is the slope of one line and m2 is the slope of another line then m1 = m2
- Two lines are perpendicular if and only if the product of their slopes is - 1, or if one is vertical and the other horizontal.
- They meet at a point to form right angles.
- They have negative reciprocal slopes.
The Product of the perpendicular lines slopes are equal to -1
i.e., m1 and m2 are slopes of two perpendicular lines and their product m1m2 = -1
i.e., m1 = -1/(m2)
|Parallel lines ||Perpendicular lines |
|y = 2x + 5 |
y = 2x - 5
y = 2x + 7
y = 2x + 115
| y = 4x + 11 |
y = -1/4x - 2
|These lines are all parallel. They all have the same slope m = 2. || These lines are perpendicular.
Their slopes (m) are negative reciprocals.
- Find the equation of the line passing through (5,4) and parallel to 3x - y - 15 = 0.
Any line parallel to 3x - y - 15 = 0 takes the form of 3x - y + k = 0.
For any equation of two parallel lines differ in constant terms, x and y coefficients being same.
If 3x - y + k = 0 passing through (5,4), we have
3.5 - 4 + k = 0
15 - 4 + k = 0
11 + k = 0
Therefore, k = -11
The required equation is 3x - y - 11 = 0.
- Find the equation of a line perpendicular to 3x - 4y + 1 = 0 and passing through (2,4).
3x - 4y + 1 = 0, Slope of this equation is m1 = -(x-coefficient / y-coefficient).
Let the slope of the line perpendicular to 3x - 4y + 1 = 0 be m2
We know that product perpendicular lines slopes are equal to -1
i.e., m1m2 = -1
We know m1 = 3/4
(3/4)m2 = -1
m2 = -4/3
The slope of line required is -4/3 and a point on the line is (2,4).
By the point slope form of a line we have,
(y - y1) = m (x - x1)
y - 4 = -4/3(x - 2)
3y - 12 = -4x + 8
4x + 3y - 20 = 0
This is the required line.
The equation of x axis or horizontal line is y = 0. The slope is 0.
The equation of y axis or vertical line is x = 0. The slope is not defined.
Directions: Answer the following questions. Also write at least 10 examples of your own.