Question

The distance between points $\left( {a + b,b + c} \right)$ and $\left( {a - b,c - b} \right)\;$ is $2\sqrt 2 b$

A) $2\sqrt {{a^2} + {b^2}}$
B) $2\sqrt {{b^2} + {c^2}}$
C) $2\sqrt 2 b$
D) $\sqrt {{a^2} - {c^2}}$

Answer 1

we have the formula to calculate distance between two points which is Distance between two points $({x_1},{y_1}) and ({x_{2,}}{y_2})$ can be calculated using the formula $\sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}}$. We already have values of $({x_1},{y_1})and({x_{2,}}{y_2})$. After putting values of $({x_1},{y_1})and({x_{2,}}{y_2})$ we can get the answer.

Complete step by step solution:
Using formula to calculate distance between two points which is Distance between two points $({x_1},{y_1})and({x_{2,}}{y_2})$
Fórmula= $\sqrt{{{({x_2} - {x_1})}^{2}}+{{({y_2} - {y_1})}^{2}}}$.
Distance between the points

& \left( {a + b,b + c} \right){\text{ }}and{\text{ }}\left( {a - b,c - b} \right) = \sqrt {{{(a - b - a - b)}^2} + {{(c - b - b - c)}^2}} \cr & = \sqrt {4{b^2} + 4{b^2}} \cr & = 2\sqrt 2 b \cr} $$ **Hence, correct answer is option C. $$2\sqrt 2 b$$** **Note:** There can be many applications of distance formula. We can show tough concepts like collinearity easily with the help of distance formula. They can give 3 points and ask whether a triangle is possible with the given vertices. In that case we know we’ll use “Sum of two sides of a triangle is always greater than the third one” but to get the distance we can use distance formula.