Equations of perpendicular and Parallel lines

Question

Parallel lines: 

1.  parallel to 2x – 3y + 11 = 0 going through (2,–1)
 

2. parallel to 3x + y = 2y – 4x + 7 going through (–2,–7)

Perpendicular lines: 

1. Perpendicular to 3y – 2x + 11 = 0 going through (2,–2) 

 

2. Find k if the lines  ""kx – 2y + 5 = 0 and 3x + 4y + 1 = 0""  are:

a. perpendicular b. parallel

Answer 1

1. parallel to 2x – 3y + 11 = 0 going through (2,–1)

2x - 3y + 11 = 0

(2/3)x + 11/3 = y

Slope = 2/3  Parallel line has the same slope

Now use point slope form to determine the equation of that line

y-2 = (2/3)(x+1)

y = (2/3)x + 2/3 + 2

y = (2/3)x +8/3

 

2. parallel to 3x + y = 2y – 4x + 7 going through (–2,–7)

3x + y = 2y - 4x + 7

7x - 7 = y

Slope = 7

y+2 = 7(x+7)

y = 7x + 49 - 2

y = 7x + 47

 

1. Perpendicular to 3y – 2x + 11 = 0 going through (2,–2)

From 1) above the slope of this line is 2/3.  So, the slope of a line perpendicular to this line is the negative inverse or -3/2

y+2 = (-3/2)(x-2)

y = (-3/2)x + 3 - 2

y = (-3/2)x + 1

 

2. Find k if the lines ""kx – 2y + 5 = 0 and 3x + 4y + 1 = 0"" are:
a. perpendicular b. parallel

3x + 4y + 1 = 0

3x+ 1 = -4y

(-3/4)x -1/4 = y

So the slope of this line is -3/4

a) And the slope of the perpendicular line (kx - 2y + 5 = 0) is 4/3

Let's work this through.

kx + 5 = 2y

(k/2)x + 5/2 = y

k/2 = 4/3

k = 8/3

b) For parallel lines, the slope is -3/4

k/2 = -3/4

k = -3/2

Answer 2

Parallel lines have same slopes.  Perpendicular lines have negative reciprocal slopes.  For each given line, convert it to  y=mx+b   form.

 

 

1)

 

-3y = -2x - 11

 

   y = (2/3)x + (11/3)

 

 

Plug in the slope and given point into the equation of the line to find b.

 

-1 = (2/3)(2) + b

 

-1 = (4/3) + b

 

-7/3 = b

 

 

The parallel line is            y = (3/2)x - (7/3)

 

 

Do the same for question 2 and question 3.  For question 3, make the slope negative reciprocal.

 

 

 

Last question

 

Convert the both lines to          y=mx+b       form.

 

4y = -3x - 1

 

  y = (-3/4)x - (1/4)

 

 

 

-2y = -kx - 5 

   y = (k / 2)x + (5 / 2)

 

 

Parallel:

 

-3 / 4 = k / 2

 

 

Solve for k.

 

 

Perpendicular:

 

k / 2 = 4 / 3

 

 

Solve for k.