Let {z₁},{z₂} be two complex numbers such that {z₁} + {z₂} and {z₁}{z₂} both are real, t

English

Gujarati

Hindi

4-1.Complex numbers

medium

Let ${z_1},{z_2}$ be two complex numbers such that ${z_1} + {z_2}$ and ${z_1}{z_2}$ both are real, then

${z_1} = - {z_2}$

${z_1} = {\bar z_2}$

${z_1} = - {\bar z_2}$

${z_1} = {z_2}$

Solution

(b) Let ${z_1} = a + ib,{z_2} = c + id$, then
${z_1} + {z_2}$ is real ==> $(a + c) + i(b + d)$is real
==> $b + d = 0$ ==> $d = – b$ …..(i)
${z_1}{z_2}$ is real ==> $(ad – bd) + i(ac + bc)$is real
==> $ad + bc = 0$ ==> $a( – b) + bc = 0$==> $a = c$
${z_1} = a + ib = c – id = {\bar z_2}$ $(\because a = c$ and $b = – d)$