DISTANCE BETWEEN PARALLEL LINES

Two lines which differs by constant is known as parallel lines. (Coefficient of "x" and "y" will be same).

General form of equations of parallel lines will be in the form as shown below.    

ax + by + c1  =  0 

ax + by + c2  =  0

Formula for the distance between two parallel lines is given by 

|c1 - c2|/√(a2 + b2)

Note : 

If the equations of the parallel lines are not given in general form, we have to convert them to general form and find the distance between them using the above formula. 

Problem 1 :

Find the distance between the following two parallel lines.

2x + 3y  =  6

2x + 3y  =  -7

Solution :

Write the equations of the parallel line in general form. 

2x + 3y - 6 = 0

2x + 3y + 7 = 0

From the above equations of parallel lines, we have

a  =  2, b  =  3, c1  =  -6 and c2  =  7

Formula for distance between parallel lines is 

=  |c1 - c2|/√(a2 + b2)

Substitute c1 = -6, c2 = 7, a = 2 and b = 3.

|-6 - 7|/√(22 + 32)

  =  |-13|/√(4 + 9)

  =  13/√13

  =  (13/√13) x (√13/√13)

  =  13√13/13

  =  √13

So, the distance between the given parallel lines is √13 units.

Problem 2 :

Find the distance between the following two parallel lines.

4x + 6y  =  -5

4x + 6y  =  -7

Solution :

Write the equations of the parallel line in general form. 

4x + 6y + 5  =  0

4x + 6y + 7  =  0

From the above equations of parallel lines, we have

a  =  4, b  =  6, c1  =  5 and c2  =  7

Formula for distance between parallel lines is 

=  |c1 - c2|/√(a2 + b2)

Substitute c1 = 5, c2 = 7, a = 4 and b = 6.

=  |5 - 7|/√(42 + 62)

  =  |-2|/√(16 + 36)

  =  2/√52

  =  2/2√13

 =  1/√13

So, the distance between the given parallel lines is 1/√13 units.

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