For any two complex number z 1- z 2 and any two real numbers a and b- - az 1- bz 2-2- bz 1- az 2-2-A- a - b - z 1-2- z 2-2B- a 2- b 2- z 1-2- z 2-2C- a 2- b 2- z 1- z 2-D- 2 a - b - z 1-2- z 2-2

We know that, |z_1 z_2|^2=|z_1|^2+|z_2|^2 2Re(z_1z_2) and |z_1+z_2|^2=|z_1|^2+|z_2|^2+2Re(z_1z_2) ∴ |az_1 bz_2|^2+|bz_1+az_2|^2=a^2|z_1|^2+b^2 |z_2|^2 2Re(abz_ ...

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Standard XII

Mathematics

Properties of Modulus

For any two c...

Question

For any two complex number $z_{1}, z_{2}$ and any two real numbers $a$ and $b$ ,
$| a z_{1} - b z_{2} |^{2} + | b z_{1} + a z_{2} |^{2} =$

(a + b) (| z_{1} |^{2} + | z_{2} |^{2})

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(a^{2} + b^{2}) (| z_{1} |^{2} + | z_{2} |^{2})

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(a^{2} + b^{2}) (| z_{1} | + | z_{2} |)

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2 (a + b) (| z_{1} |^{2} + | z_{2} |^{2})

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Solution

The correct option is B $(a^{2} + b^{2}) (| z_{1} |^{2} + | z_{2} |^{2})$
We know that,
$[| z_{1} - z_{2} |^{2} = | z_{1} |^{2} + | z_{2} |^{2} - 2 R e (z_{1}_{2})]$
and
$[| z_{1} + z_{2} |^{2} = | z_{1} |^{2} + | z_{2} |^{2} + 2 R e (z_{1}_{2})]$
$∴ | a z_{1} - b z_{2} |^{2} + | b z_{1} + a z_{2} |^{2} = a^{2} | z_{1} |^{2} + b^{2} | z_{2} | - 2 R e (a b z_{1}_{2}) + b^{2} | z_{1} |^{2} + a^{2} | z_{2} | + 2 R e (a b z_{1}_{2})$
$= | z_{1} |^{2} (a^{2} + b^{2}) + | z_{2} |^{2} (a^{2} + b^{2})$
$= (a^{2} + b^{2}) (| z_{1} |^{2} + | z_{2} |^{2})$

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For any two complex numbers $z_{1} z_{2}$ and any two real numbers a and b $| a z_{1} - b z_{2} |^{2} + | b z_{1} + a z_{2} |^{2}$ =

Q. For any two complex numbers

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and

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any real numbers and

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

Q. For any two complex numbers

z_{1}, z_{2}

and any two real numbers

a, b

show that

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

Q. 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.

State True and False

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}] .

Type 1 for true and 0 for false

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Properties of Modulus

Standard XII Mathematics

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For any two complex number z1,z2 and any two real numbers a and b, |az1−bz2|2+|bz1+az2|2=

For any two complex number $z_{1}, z_{2}$ and any two real numbers $a$ and $b$ ,
$| a z_{1} - b z_{2} |^{2} + | b z_{1} + a z_{2} |^{2} =$