InterviewSolution
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InterviewSolution
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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 ofcomplex nunmbers `omega=a+ic and omega_2=b+id` satisfiesA. `|omega_(1)|=1`B. `|omega_(2)|=1`C. `Re(omega_(1)baromega_(2)) = 0`D. `Im(omega_(1)baromega_(2))=0` |
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Answer» Correct Answer - A::B::C `|Z_(1)|=|Z_(2)| = 1` `rArr a^(2) + b^(2) =c^(2) + a^(2) = 1" "(1)` and `Re(z_(1)barz_(2)) =0` `rArr Re{(a+ib)(c+id)} = 0` `rArrac + bd = 0" "(2)` Now from Eqs. (1) and (2) we get `a^(2) + b^(2) =1` ` rArr a^(2) + (a^(2)c^(2))/(d^(2)) = 1` `rArr a^(2) =d^(2)" "(3)` Also, `c^(2)+d^(2) =1` `rArr c^(2) +(a^(2)+c^(2))/(b^(2))=1` `rArr b^(2) = c^(2)` `|omega_(1)| = sqrt(a^(2) + b^(2)) = sqrt(a^(2) +b^(2)) = 1` [From (1) and (4)] `and |omega_(2)|=sqrt(b^(2) + d^(2))= sqrt(c^(2) + d^(2))=1`[From (1) and (4)] Further `Re(omega_(1)baromega_(2)) = Re{(a+ ic)(b-id)}` `ab+ cd` `=ab -(ac^(2))/(b)" "["From (2)"]` `= (ab^(2) - ac^(2))/(b) =0" "["Form (4)"]` Also, `Im(omega_(1)baromega_(2)) = bc-ad` `= bc-a(-(ac)/(b)) = ((a^(2) +b^(2))c)/(b) = (c)/(b) = pm1ne 0` `therefore |oemga_(1)|= 1, |omega_(2)|=1 and Re(omega_(1)baromega_(2))=0` |
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