If ` z_(1) = a + ib " and " z_(2) + c id ` are complex numbers such that ` |z_(1)| = |z_(2)| = 1 ` and ` Re (z_(1)bar (z)_(2)) = 0 `,
then the pair of complex numbers ` w_(1) = a + ic " and " w_(2) = b id ` satisfies :

Explanation

we have ,
`|z_(1)|=|z_(2)|=1 `
` Rightarrow p^(2)+q^(2) = u^(2)+v^(2)=1`
` and Re(z_(1)barz_(2))=0`
` Rightarrow Re(p+iq) ( u-iv) =0`
` Rightarrow pu + qv =0`
` Rightarrow p/q = - v/u = k , say`
`Rightarrow (k^(2)+1)q^(2)= (k^(2)+1)u^(2)`
` Rightarrow q^(2)=u^(2)or (1/k^(2)+1)p^(2) ( 1/k^(2) +1) v^(2) and p^(2) = v^(2)`
Now `|omega_(1)|=sqrt(p^(2)+q^(2))=sqrt(u^(2)+v^(2))=1`
` |omega_(2)|=sqrt(u^(2)+v^(2))=sqrt(q^(2)+p^(2))=1`
`Re (omega_(1) baromega_(2)) = Re (bar(q+iv))( p +iu) = pq + uv = ( qk)k - k ( uk)`
`= (q^(2)-u^(2))k^(2)=0`
Hence, (1), (2), (3) are correct options.