DISTANCE BETWEEN PARALLEL LINES

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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