Question

is cu 1. Find the distance between the following pairs of points:
(i) (-5,7).(-1,3) (ii) (a, b),(-a, -b)


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Answer 1

$\underline { \pink { Distance \: formula : }}$

$The \:distance \: between \:two \: given \\points \: A(x_{1} ,y_{1}) , \: B(x_{2} ,y_{2}) \: is \\AB = \sqrt{ (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} }$

$i) Here, A(x_{1} ,y_{1}) = ( -5,7) \: and \\B(x_{2} ,y_{2}) = ( -1 , 3 )$

$AB = \sqrt{ [-1 -(-5)]^{2} + (3-7)^{2} }\\= \sqrt{ (-1+5)^{2} + (-4)^{2} } \\= \sqrt{ 4^{2} + 4^{2} } \\= \sqrt{ 16 + 16 } \\= \sqrt{ 32 } \\= 4\sqrt{2}$

$\red{\therefore Distance } \green { = 4\sqrt{2}}$

$ii) Here, A(x_{1} ,y_{1}) = ( a , b ) \: and \\B(x_{2} ,y_{2}) = ( -a , -b )$

$AB = \sqrt{ (-a - a)^{2} + (-b-b)^{2} }\\= \sqrt{ (-2a)^{2} + (-2b)^{2} } \\= \sqrt{ 4a^{2} + 4b^{2} } \\= \sqrt{ 4(a^{2} + b^{2} ) } \\= 2\sqrt{(a^{2} + b^{2})}$

$\red{\therefore Distance } \green { =2\sqrt{(a^{2} + b^{2})}}$

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