| z_1 + z_2 |^2 + | z_1 - z_2 |^2 is equal to - Vidyadip

MCQ

${\left| {{z_1} + {z_2}} \right|^2} + {\left| {{z_1} - {z_2}} \right|^2}$ is equal to

A
$2\left( {\left| {{z_1}} \right| + \left| {{z_2}} \right|} \right)$
✓
$2\left( {{{\left| {{z_1}} \right|}^2} + {{\left| {{z_2}} \right|}^2}} \right)$
C
$\left| {{z_1}} \right|\left| {{z_2}} \right|$
D
${{{\left| {{z_1}} \right|}^2} + {{\left| {{z_2}} \right|}^2}}$

✓

Answer

Correct option: B.

$2\left( {{{\left| {{z_1}} \right|}^2} + {{\left| {{z_2}} \right|}^2}} \right)$

b
${\therefore \quad {{\left| {{z_1} + {z_2}} \right|}^2} + {{\left| {{z_1} - {z_2}} \right|}^2}}$

${ = {{\left| {{z_1}} \right|}^2} + {{\left| {{z_2}} \right|}^2} + }$ ${2\left| {{z_1}} \right|\left| {{z_2}} \right| + {{\left| {{z_1}} \right|}^2}}$ ${ + {{\left| {{z_2}} \right|}^2} - 2\left| {{z_1}} \right|\left| {{z_2}} \right|}$

${ = 2{{\left| {{z_1}} \right|}^2} + 2{{\left| {{z_2}} \right|}^2}}$ ${ = 2\left[ {{{\left| {{z_1}} \right|}^2} + {{\left| {{z_2}} \right|}^2}} \right]}$

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