For any two complex numbers z1- z2 and any real numbers a and b- -az1 - bz2-2 - -bz1 - az2-2 -a2 - b2-z1-2 - -z2-2- If this is true enter 1- else enter 0-

|az_ 1 bz_ 2 |^ 2 +|bz_ 1 +az_ 2 |^ 2 =( a z _ 1 b z _ 2 ) ( az̅ ̅_̅ ̅1̅ bz̅ ̅_̅ ̅2̅ #160;) +( a z _ 1 +b z _ 2 ) ( az̅ ̅_̅ ̅1̅ +bz̅ ̅_̅ ̅2̅ #160;) ...

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Byju's Answer

Standard XII

Mathematics

Properties of Modulus

For any two c...

Question

For any two complex numbers $z_{1}, z_{2}$ and any real numbers a and b, ${| a z_{1} - b z_{2} |}_{1}^{2} + {| b z_{1} + a z_{2} |}_{1}^{2} = (a^{2} + b^{2}) ({| z_{1} |}_{1}^{2} + {| z_{2} |}_{2}^{2})$ . If this is true enter 1, else enter 0.

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Solution

$| a z_{1} - b z_{2} |^{2} + | b z_{1} + a z_{2} |^{2} = (a z_{1} - b z_{2}) (a_{1} - b_{2}) + (a z_{1} + b z_{2}) (a_{1} + b_{2})$

$= a^{2} {| z_{1} |}^{2} - a b z_{1}_{2} - a b z_{2}_{1} + b^{2} {| z_{2} |}^{2} + a^{2} {| z_{1} |}^{2} + a b z_{1}_{2} + a b z_{2}_{2} + b^{2} {| z_{2} |}^{2}$

$\Rightarrow | a z_{1} - b z_{2} |^{2} + | b z_{1} + a z_{2} |^{2} = (a^{2} + b^{2}) (| z_{1} |^{2} + | z_{2} |^{2})$
Ans: 1

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Q. For any two complex numbers

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and any two real numbers

a, b

show that

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State True and False

For any two complex numbers

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Type 1 for true and 0 for false

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For any two complex numbers z1,z2 and any real numbers a and b, |az1−bz2|2+|bz1+az2|2=(a2+b2)(|z1|2+|z2|2). If this is true enter 1, else enter 0.

|az1−bz2|2+|bz1+az2|2=(az1−bz2)(a¯z1−b¯z2)+(az1+bz2)(a¯z1+b¯z2)=a2|z1|2−abz1¯z2−abz2¯z1+b2|z2|2+a2|z1|2+abz1¯z2+abz2¯z2+b2|z2|2⇒|az1−bz2|2+|bz1+az2|2=(a2+b2)(|z1|2+|z2|2)Ans: 1

For any two complex numbers $z_{1}, z_{2}$ and any real numbers a and b, ${| a z_{1} - b z_{2} |}_{1}^{2} + {| b z_{1} + a z_{2} |}_{1}^{2} = (a^{2} + b^{2}) ({| z_{1} |}_{1}^{2} + {| z_{2} |}_{2}^{2})$ . If this is true enter 1, else enter 0.