The distance between points \[\left( {a + b,b + c} \right)\]and \[\left( {a - b,c - b} \right)\]is:

a. \[2\sqrt {{a^2} + {b^2}} \]
b. \[2\sqrt {{b^2} + {c^2}} \]
c. \[2\sqrt 2 b\]
d. \[\sqrt {{a^2} - {c^2}} \]

Hint: Here in this given problem we are to find the distance between two given points. We use a formula of distance between two points, and then analyze which option is going well with the answer.

Complete step-by-step answer:
We need to find the distance between two given points,
So, now
Distance between two points \[\left( {{x_{1}},{y_1}} \right)\] and \[\left( {{x_{2}},{y_2}} \right)\]can be calculated, using the formula,
 \[\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]
Thus, Distance between the points \[\left( {a + b,b + c} \right)\] and \[\left( {a - b,c - b} \right)\] is,
\[\sqrt {{{\left( {a - b - a - b} \right)}^2} + {{\left( {c - b - b - c} \right)}^2}} \]
On simplifying the bracket we get,
\[ = \sqrt {{{\left( { - 2b} \right)}^2} + {{\left( { - 2b} \right)}^2}} \]
On squaring we get,
\[ = \sqrt {4{b^2} + 4{b^2}} \]
On adding terms we get,
\[ = \sqrt {8{b^2}} \]
On taking square root, we get
\[ = 2\sqrt {2} b\]units
So, the answer of our problem is, option c, \[ = 2\sqrt {2} b\]units.

Note: The formula we have used in this problem is the formula between two points, \[\left( {{x_{1}},{y_1}} \right)\]and \[\left( {{x_{2}},{y_2}} \right)\]is denoted by, \[\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \] . We need to keep in mind that we can interchange the 1st point and 2nd point among themselves as it is a scalar quantity which is not dependent on directions.