FINDING EQUATIONS OF PARALLEL AND PERPENDICULAR LINES

Example 1 :

A straight line passes through the points A(1, 4) and B(5, 16)

a) Find the equation of the line parallel to AB that passes through (1, 7)

b)  Find the equation of the line perpendicular to AB that passes through the midpoint of AB.

Solution :

a)

Given points, A(1, 4) and B(5, 16).

A(1, 4)----->(x₁, y₁)

B(5, 16)----->(x₂, y₂)

Equation of line AB is 2x - y + 2 = 0.

Equation of line parallel to AB is 2x - y + k = 0.

The line passing through (1, 7).

2x - y + k = 0

2(1) - 7 + k = 0

2 - 7 + k = 0

-5 + k = 0

k = 5

So, the equation of line parallel to AB is 2x - y + 5 = 0.

b)

Given points, A(1, 4) and B(5, 16).

A(1, 4)----->(x₁, y₁)

B(5, 16)----->(x₂, y₂)

Midpoint of AB = [(x₁ + x₂)/2, (y₁ + y₂)/2]

= [(1 + 5)/2, (4 + 16)/2]

= [6/2, 20/2]

Midpoint of AB = (3, 10)

To find the equation of line perpendicular to AB,

So, the equation of the line is x + 2y - 23 = 0.

Example 2 :

Are the lines 2x + y = 8 and y = 2x + 5 parallel?

Solution :

Line 1 ----> 2x + y = 8

y = -2x + 8 ----(1)

Comparing y = mx + c in (1), we get

m₁ = -2

Line 2 ----> y = 2x + 5

y = 2x + 5 -----(2)

m₂ = 2

m₁ ≠ m₂

So, the lines are not parallel.

Example 3 :

Are the lines 4x - y - 5 = 0 and x + 4y + 1 = 0 perpendicular?

Solution :

Line 1 ----> 4x - y - 5 = 0

-y = -4x + 5

y = 4x - 5 ----(1)

Comparing y = mx + c in (1), we get

m₁ = 4

Line 2 ----> x + 4y + 1 = 0

4y = -x - 1

y = -x/4 - 1/4 -----(2)

m₂ = -1/4

m₁ . m₂ = -1

4 . -1/4 = -1

-1 = -1

So, the lines are perpendicular.

Example 4 :

The Line L has equation y = 2x + 8.

The line L crosses the x-axis at the point A.

The line M is perpendicular to Line L and passes through the point A

a) Find the coordinates of the point A.

b) Find equation of the line M.

Solution :

a)

 y = 2x + 8

Find the x-coordinates at the point A:

when y = 0

0 = 2x + 8

-2x = 8

x = -4

So, the coordinates of the Point A is (-4, 0).

b)

The equation of the line L = y = 2x + 8

So, the equation of the line is y = -x/2 - 2.

Example 5 :

The point A has coordinates (-12, -7) and the point B has coordinates (-8, 1). Find the equation of the line parallel to AB and passing through (2, 5)

Solution :

Given points, A(-12, -7) and B(-8, 1).

A(-12, -7)----->(x₁, y₁)

B(-8, 1)----->(x₂, y₂)

Equation of line AB is 2x - y + 17 = 0.

Equation of line parallel to AB is 2x - y + k = 0.

The line passing through (2, 5).

2x - y + k = 0

2(2) - 5 + k = 0

4 - 5 + k = 0

-1 + k = 0

k = 1

So, the equation of line parallel to AB is 2x - y + 1 = 0.

Example 6 :

The line L passes through the points (-2, 1) and (2, 3)

The line N passes through the points (4, 7) and (12, 11)

Bryan says that the lines L and N are parallel.

Is Bryan correct? Explain your answer?

Solution :

Since the slopes are not equal, they are not parallel.