Question

Find the distance between the points $R\left( {a + b,\,a - b} \right)\,and\,\,S\left( {a - b,\, - a - b} \right)$.

Answer 1

Hint: Firstly we will convert the given points in the form of $A\left( {{x_1},{y_1}} \right)$ and $B\left( {{x_2},{y_2}} \right)$ . So,${x_1} = a + b,{y_1} = a - b$ and ${x_2} = a - b,{y_2} = - a - b$.Thereafter we will substitute the value of ${x_1},{y_1},{x_2},{y_2}$in the distance formula. By using distance formula $\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $


Complete step by step solution:
$R\left( {a + b,\,a - b} \right)$
$S\left( {a - b,\, - a - b} \right)$
  {$x_1$} = a + b & {$y_1$} = a - b
  {$x_2$} = a - b & {$y_2$}= - a - b
Distance formula: $\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $ …(i)
Substitute the value of x, $x_2$, $y_1$ and $y_2$ in equation (i)
$RS = \sqrt {{{\left( {\left( {a - b} \right) - \left( {a + b} \right)} \right)}^2} + {{\left( {\left( { - a - b} \right) - \left( {a - b} \right)} \right)}^2}} $
$ = \sqrt {{{\left( {a - b - a - b} \right)}^2} + {{\left( { - a - b - a + b} \right)}^2}} $
$ = \sqrt {{{\left( { - 2b} \right)}^2} + {{\left( { - 2a} \right)}^2}} $
$ = \sqrt {4{b^2} + 4{a^2}} $
$ = 2\sqrt {{b^2} + {a^2}} $ ans.


Note: Students should solve the problem carefully and put the exact values of ${x_1},{y_1},{x_2},{y_2}$ in the distance formula.If you will make a mistake somewhere then you will get wrong answer.