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

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

Explanation

Solution

Firstly we will convert the given points in the form of A(x1,y1)A\left( {{x_1},{y_1}} \right) and B(x2,y2)B\left( {{x_2},{y_2}} \right) . So,x1=a+b,y1=a−b{x_1} = a + b,{y_1} = a - b and x2=a−b,y2=−a−b{x_2} = a - b,{y_2} = - a - b.Thereafter we will substitute the value of x1,y1,x2,y2{x_1},{y_1},{x_2},{y_2}in the distance formula. By using distance formula (x2−x1)2+(y2−y1)2\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}}

Complete step by step solution:
R(a+b, a−b)R\left( {a + b,\,a - b} \right)
S(a−b, −a−b)S\left( {a - b,\, - a - b} \right)
{x1x_1} = a + b & {y1y_1} = a - b
{x2x_2} = a - b & {y2y_2}= - a - b
Distance formula: (x2−x1)2+(y2−y1)2\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} …(i)
Substitute the value of x, x2x_2, y1y_1 and y2y_2 in equation (i)
RS=((a−b)−(a+b))2+((−a−b)−(a−b))2RS = \sqrt {{{\left( {\left( {a - b} \right) - \left( {a + b} \right)} \right)}^2} + {{\left( {\left( { - a - b} \right) - \left( {a - b} \right)} \right)}^2}}
=(a−b−a−b)2+(−a−b−a+b)2= \sqrt {{{\left( {a - b - a - b} \right)}^2} + {{\left( { - a - b - a + b} \right)}^2}}
=(−2b)2+(−2a)2= \sqrt {{{\left( { - 2b} \right)}^2} + {{\left( { - 2a} \right)}^2}}
=4b2+4a2= \sqrt {4{b^2} + 4{a^2}}
=2b2+a2= 2\sqrt {{b^2} + {a^2}} ans.

Note: Students should solve the problem carefully and put the exact values of x1,y1,x2,y2{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.