Question

The distance between the lines 2x + 4 =0 and x -5 = 0 is:-----

Answer 1

The distance between the lines 2x + 4 =0 and x -5 = 0 is 7 units.

Answer 2

Given:

Two straight lines 2x + 4 = 0 and x - 5 = 0.

To Find:

The distance between the two given lines.

Solution:

The given problem can be solved by using the concepts of straight lines.

1. The given lines are 2x + 4 = 0 ( Consider it as Equation 1 ) and x - 5 = 0 (Consider it as Equation 2).

2. Equation 1 can be further modified as,

=> 2x + 4 = 0 (Divide the equation by 2 on both the sides).

=> x + 2 = 0. ( Consider it as Equation 3)

3. According to the properties of straight lines,

• Let ax + by + c =0 be a line, Two lines are said to be parallel when the coefficients of x and y are the same and they only differ in the value of the constant.

Example: ax + by + c = 0 and ax + by + d = 0 are parallel lines since they differ only in the constant value.

4. Equation 3 and 4 are x + 2 = 0 and x - 5 = 0, the two equations has the same coefficient but different constant value. Hence the two lines are parallel to each other.

5. According to the properties of straight lines,

• The distance between two parallel lines  ax + by + c = 0 and ax + by + d = 0 is given by the relation  $\frac{|c-d|}{\sqrt{a^{2}+b^{2} } }$. where | | is the modulus of a number.

6. On Substituting the values in the above formula we get,

=> Distance between the lines =$\frac{|-5 -2|}{\sqrt{1^{2} +0^{2} } }$,

=> Distance between the lines = 7 units.

Therefore, The distance between the given lines is 7 units.