Answer

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Hint – Whenever we are given two points, we can easily find the distance between them using the concept of distance formula. Use this same concept to find distance for all three options.

Whenever we are given two pair of point such that the points are $\left( {{x_1},{y_1}} \right){\text{ and }}\left( {{x_2},{y_2}} \right)$ then the distance between these points can be given as $\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $……………. (1)

Now the first pair of point is $\left( i \right){\text{ }}\left( {2,3} \right),\left( {4,1} \right)$

Using equation (1) we get

Distance = $\sqrt {{{\left( {4 - 2} \right)}^2} + {{\left( {1 - 3} \right)}^2}} $

Distance= $\sqrt {{2^2} + {{\left( { - 2} \right)}^2}} $

$ \Rightarrow \sqrt {4 + 4} = 2\sqrt 2 $ m

Now the second pair of point is $\left( {ii} \right){\text{ }}\left( { - 5,7} \right),\left( { - 1,3} \right)$

Using equation (1) we get

Distance = $\sqrt {{{\left( { - 1 - \left( { - 5} \right)} \right)}^2} + {{\left( {3 - 7} \right)}^2}} $

Distance= $\sqrt {{4^2} + {{\left( { - 4} \right)}^2}} $

$ \Rightarrow \sqrt {16 + 16} = 4\sqrt 2 $ m

Now the second pair of point is $\left( {iii} \right){\text{ }}\left( {a,b} \right),\left( { - a, - b} \right)$

Using equation (1) we get

Distance = $\sqrt {{{\left( { - a - \left( a \right)} \right)}^2} + {{\left( { - b - b} \right)}^2}} $

Distance= $\sqrt {{{\left( { - 2a} \right)}^2} + {{\left( { - 2b} \right)}^2}} $

$ \Rightarrow \sqrt {4{a^2} + 4{b^2}} = 2\sqrt {{a^2} + {b^2}} $ m

Note – Whenever we face such type of problems, the key concept is compare the given points whose distance is to be calculated with any general point $\left( {{x_1},{y_1}} \right){\text{ and }}\left( {{x_2},{y_2}} \right)$ and then applying the distance formula to obtain the right distance between them.

Whenever we are given two pair of point such that the points are $\left( {{x_1},{y_1}} \right){\text{ and }}\left( {{x_2},{y_2}} \right)$ then the distance between these points can be given as $\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $……………. (1)

Now the first pair of point is $\left( i \right){\text{ }}\left( {2,3} \right),\left( {4,1} \right)$

Using equation (1) we get

Distance = $\sqrt {{{\left( {4 - 2} \right)}^2} + {{\left( {1 - 3} \right)}^2}} $

Distance= $\sqrt {{2^2} + {{\left( { - 2} \right)}^2}} $

$ \Rightarrow \sqrt {4 + 4} = 2\sqrt 2 $ m

Now the second pair of point is $\left( {ii} \right){\text{ }}\left( { - 5,7} \right),\left( { - 1,3} \right)$

Using equation (1) we get

Distance = $\sqrt {{{\left( { - 1 - \left( { - 5} \right)} \right)}^2} + {{\left( {3 - 7} \right)}^2}} $

Distance= $\sqrt {{4^2} + {{\left( { - 4} \right)}^2}} $

$ \Rightarrow \sqrt {16 + 16} = 4\sqrt 2 $ m

Now the second pair of point is $\left( {iii} \right){\text{ }}\left( {a,b} \right),\left( { - a, - b} \right)$

Using equation (1) we get

Distance = $\sqrt {{{\left( { - a - \left( a \right)} \right)}^2} + {{\left( { - b - b} \right)}^2}} $

Distance= $\sqrt {{{\left( { - 2a} \right)}^2} + {{\left( { - 2b} \right)}^2}} $

$ \Rightarrow \sqrt {4{a^2} + 4{b^2}} = 2\sqrt {{a^2} + {b^2}} $ m

Note – Whenever we face such type of problems, the key concept is compare the given points whose distance is to be calculated with any general point $\left( {{x_1},{y_1}} \right){\text{ and }}\left( {{x_2},{y_2}} \right)$ and then applying the distance formula to obtain the right distance between them.

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