Question

What is the distance between the parallel lines y = x + a and y = x + b?
(a). \[\dfrac{{|a - b|}}{{\sqrt 2 }}\]
(b). \[|a - b|\]
(c). \[|a + b|\]
(d). \[\dfrac{{|a + b|}}{{\sqrt 2 }}\]

Answer 1

Hint: The distance between two parallel lines \[ax + by + {c_1} = 0\] and \[ax + by + {c_2} = 0\] is given by the formula \[d = \dfrac{{|{c_1} - {c_2}|}}{{\sqrt {{a^2} + {b^2}} }}\]. Use this formula to find the distance between the given lines.

Complete step-by-step answer:
Two lines are said to be parallel if they do not intersect at any finite point in the space. They always maintain the same distance between them.
The equations of the parallel lines have the x and y coefficient as proportional to each other.
For finding the distance between the two parallel lines, we first express the two equations such that the coefficients of x and y are equal.
The equations y = x + a and y = x + b already have their x and y coefficients equal.
We express them in the standard form ax + by + c = 0
\[x - y + a = 0...............(1)\]
\[x - y + b = 0..............(2)\]
Now, we use the formula for calculating the distance between two parallel lines \[ax + by + {c_1} = 0\] and \[ax + by + {c_2} = 0\] given as follows:
\[d = \dfrac{{|{c_1} - {c_2}|}}{{\sqrt {{a^2} + {b^2}} }}\]
The distance between the two parallel lines in equation (1) and equation (2) is given as follows:
\[d = \dfrac{{|a - b|}}{{\sqrt {{1^2} + {1^2}} }}\]
Simplifying the above equation, we have as follows:
\[d = \dfrac{{|a - b|}}{{\sqrt {1 + 1} }}\]
\[d = \dfrac{{|a - b|}}{{\sqrt 2 }}\]
Hence, the correct answer is the option (a).

Note: You may forget the factor of \[\dfrac{1}{{\sqrt 2 }}\] in the final answer and may get the option (b) which is wrong. You might also forget the negative sign in the formula \[d = \dfrac{{|{c_1} - {c_2}|}}{{\sqrt {{a^2} + {b^2}} }}\] and may arrive at option (d), which is again a wrong answer.