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Find the distance between $A\left( {a + b,b - a} \right)$ and $B\left( {a - b,a + b} \right)$?

Last updated date: 20th Jun 2024
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Hint: In this question we will use distance formula to find the distance between two points. Distance formula, $d = \sqrt {{{\left( {{x_1} - {x_2}} \right)}^2} + {{\left( {{y_1} - {y_2}} \right)}^2}} $.

Complete step-by-step answer:
Now points given are,
$A\left( {a + b,b - a} \right)$ and $B\left( {a - b,a + b} \right)$
Distance Formula: The distance formula is used to determine the distance, $d$ , between two points. If the coordinates of the two points are $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$, the distance equals the square root of $\left( {{x_1} - {x_2}} \right)$ squared $ + \left( {{y_1} - {y_2}} \right)$ squared.
$d = \sqrt {{{\left( {{x_1} - {x_2}} \right)}^2} + {{\left( {{y_1} - {y_2}} \right)}^2}} $
Now, the distance between $A$ and$B$ is determined by using the distance formula.
   \Rightarrow {\text{ }}AB = \sqrt {{{\left( {\left( {a + b} \right) - \left( {a - b} \right)} \right)}^2} + {{\left( {\left( {b - a} \right) - \left( {a + b} \right)} \right)}^2}} \\
  {\text{or }}AB = \sqrt {{{\left( {a + b - a + b} \right)}^2} + {{\left( {b - a - a - b} \right)}^2}} \\
  {\text{or }}AB = \sqrt {{{\left( {2b} \right)}^2} + {{\left( { - 2a} \right)}^2}} \\
  {\text{or }}AB = \sqrt {4{b^2} + 4{a^2}} \\
  {\text{or }}AB = \sqrt {4\left( {{a^2} + {b^2}} \right)} \\
  {\text{or }}AB = 2\sqrt {\left( {{a^2} + {b^2}} \right)} \\
Thus the distance between $A\left( {a + b,b - a} \right)$ and $B\left( {a - b,a + b} \right)$$ = 2\sqrt {\left( {{a^2} + {b^2}} \right)} $

Note: These types of questions can be solved by using distance formulas. In this question, we simply apply the distance formula between the given point and we get the distance between the points $A{\text{ and }}B$.