If `z_1=a_ib and z_2=c+id` are complex numbes such that `|z_1|=|z_2|=1 and Re(z_1barz_2)=0` then the pair of complex numbers `omega_1= a+ic and omega_2=b+id` satisfy which of the following relations? (A) `|omega_1|=1` (B) `|omega_2|=1` (C) `Re(omega_1 baromega_2)=0` (D) `Im(omega_1baromega_2)=0`

Options

A. `|w_1|=1`
B. `|w_2|=1`
C. `Re(w_1barw_2)=0`
D. None of these

Answer

A, B, C

Explanation

Since `z_1=a+ib " and " z_2=c+id `
`rArr |z_1|^2=a^2+b^2=1" and "|z_2|^2=c^2+d^2=1`
`[because |z_1|=|z_2|=1]`
Also Re `(z_1 bar z_2)=0 rArr ac+bd =0`
`rArr a/b=-d/c=lambda`
From Eqs.(i) and (ii) `b^2lambda^2+b^2=c^2+lambda^2c^2`
`rArr b^2=c^2 and a^2 =d^2`
Also ,given `w_1 = a+ ic and w_2 =b+id `
Now `|w_1|=sqrt(a^2+c^2)=sqrt(a^2+b^2)=1`
`|w_2|=sqrt(b^2+d^2)=sqrt(a^2+b^2)=1`
and `Re(w_1 barw_2)=ab+cd=(b lambda)b+c (-lambdac)` [ from Eq.(i)]
`=lambda(b^2-c^2)`