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If z and omega are two non-zero complex ...

If z and `omega` are two non-zero complex numbers such that `|z omega|=1" and "arg(z)-arg(omega)=(pi)/(2)`, then `bar(z)omega` is equal to

A

`bar z w =i`

B

`zbarw=(1-i)/sqrt(2)`

C

`bar z w=i`

D

`z bar w = (-1 +i)/sqrt(2)`

Text Solution

Verified by Experts

The correct Answer is:
A
It is given that there, are two complex numbers z and w sunc that `|z w|=1 and arg(w) = pi//2`
`therefore |z||w|=1 [ because |z_1 z_2|= | z_1||z_2|]`
and `arg(z) = pi/2 + arg (w)`
Let `|z| =r , then | w|= 1/r`
and let arg (w) `= theta " then arg(z) pi/2 + theta `
So we can assume
`z= re^(i pi//2+ theta)`
`[ because ` if z= x+ iy is a complex number then it can be written as z `=re^(i theta) where r= |z| and theta = arg (z) ]`
and `w =1/re^(it theta)`
Now ` barz.w = re^(-i(pi//2theta)). 1/re^(i theta)`
`=e^(i(-pi//2- theta+ theta))=e^(-i(pi//2))-i " " [ because e^(-i theta) = cos theta - i sin theta]`
and `z bar w = re^(i(pi//2 + theta)).1/r e^(- i theta)`
`= e^(i(pi//2+ theta - theta))=e^(i(pi//2))=i`
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