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If $Z_1$ and $Z_2$ are conjugate complex numbers. Match the items under the following columns?
Options :
$\begin{array}{llll}A & B & C & D\end{array}$
Options:
Options :
$\begin{array}{llll}A & B & C & D\end{array}$
Solution:
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Verified Answer
The correct answer is:
$\begin{array}{llll}3 & 1 & 2 & 4\end{array}$
Given $Z_1, Z_2$ are conjugate complex numbers.
Let $Z_1=a+i b$
$\Rightarrow \quad\left|Z_1\right|^2=\left(\sqrt{a^2+b^2}\right)^2$
$=a^2+b^2$
$\Rightarrow \quad Z_2=a-i b$
(i) $Z_1 Z_2=(a+i b)(a-i b)$
$=a^2+b^2=\left|Z_1\right|^2$
(ii) $Z_1+Z_2=0$
$a+i b+a-i b=0$
$2 a=0 \Rightarrow a=0$
$\Rightarrow$ imaginary axis.
(iii) If $Z_1=a+i b$
$\begin{aligned} & \operatorname{Im}\left(Z_1\right)=b \\ & \operatorname{Im}\left(-Z_2\right)=\operatorname{Im}(-(a-i b))\end{aligned}$
$=\operatorname{Im}(-a+i b)=b$
Hence, $\operatorname{Im}\left(Z_1\right)=\operatorname{Im}\left(-Z_2\right)$
(iv) $\operatorname{Re}\left(Z_1\right)=a$
$\operatorname{Re}\left(Z_2\right)=a$
$\Rightarrow \operatorname{Re}\left(Z_1\right)=\operatorname{Re}\left(Z_2\right)$
$\therefore \mathrm{A}-(3) ; \mathrm{B}-(1) ; \mathrm{C}-(2) ; \mathrm{D}-(4)$.
Let $Z_1=a+i b$
$\Rightarrow \quad\left|Z_1\right|^2=\left(\sqrt{a^2+b^2}\right)^2$
$=a^2+b^2$
$\Rightarrow \quad Z_2=a-i b$
(i) $Z_1 Z_2=(a+i b)(a-i b)$
$=a^2+b^2=\left|Z_1\right|^2$
(ii) $Z_1+Z_2=0$
$a+i b+a-i b=0$
$2 a=0 \Rightarrow a=0$
$\Rightarrow$ imaginary axis.
(iii) If $Z_1=a+i b$
$\begin{aligned} & \operatorname{Im}\left(Z_1\right)=b \\ & \operatorname{Im}\left(-Z_2\right)=\operatorname{Im}(-(a-i b))\end{aligned}$
$=\operatorname{Im}(-a+i b)=b$
Hence, $\operatorname{Im}\left(Z_1\right)=\operatorname{Im}\left(-Z_2\right)$
(iv) $\operatorname{Re}\left(Z_1\right)=a$
$\operatorname{Re}\left(Z_2\right)=a$
$\Rightarrow \operatorname{Re}\left(Z_1\right)=\operatorname{Re}\left(Z_2\right)$
$\therefore \mathrm{A}-(3) ; \mathrm{B}-(1) ; \mathrm{C}-(2) ; \mathrm{D}-(4)$.
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