State True and FalseFor 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 - Type 1 for true and 0 for false

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

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Properties of Conjugate of a Complex Number

State True an...

Question

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

Open in App

Solution

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

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

$= (a^{2} + b^{2}) [{| z_{1} |}_{1}^{2} + {| z_{2} |}_{2}^{2}]$

Suggest Corrections

Similar questions

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.

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 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} =

Q. For any two complex numbers

z_{1}

and

z_{2}

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 z₁, z₂ and any real numbers a, b, |az₁ – bz₂|² + |bz₁ + az₂|² = ____________.

Join BYJU'S Learning Program

State True and FalseFor 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]. Type 1 for true and 0 for false

|az1−bz2|2+|bz1+az2|2=(az1−bz2)(a¯z1−b¯z2)+(az1+bz2)(a¯z1+b¯z2)=a2|z1|2+b2|z2|2−abz1¯z2−abz2¯z2+a2|z1|2+b2|z2|2+abz1¯z2+abz2¯z2=(a2+b2)[|z1|2+|z2|2]

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