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If the point $z_{1}=1+i$ where $i=\sqrt{-1}$ is the reflection of a point $\mathrm{z}_{2}=\mathrm{x}+$ iy in the line $\mathrm{i} \overline{\mathrm{z}}-\mathrm{i} \mathrm{z}=5$, then the point $\mathrm{z}_{2}$ is
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The correct answer is:
$\quad 1+4 \mathrm{i}$
Let $z=a+b i$ 74
$\Rightarrow \bar{z}=\mathrm{a}-\mathrm{bi}$
$\therefore \mathrm{i} \overline{\mathrm{z}}-\mathrm{iz}=\mathrm{i}[(\mathrm{a}-\mathrm{bi})-(\mathrm{a}+\mathrm{bi})]=5$
$\Rightarrow \mathrm{i}[-2 \mathrm{bi}]=5$
$\Rightarrow \mathrm{b}=\frac{5}{2}$
So from figure it is clear that
$x=1, y=\frac{5}{2}+\frac{3}{2}=4$
$\mathrm{z}_{2}=1+4 \mathrm{i}$
$\Rightarrow \bar{z}=\mathrm{a}-\mathrm{bi}$
$\therefore \mathrm{i} \overline{\mathrm{z}}-\mathrm{iz}=\mathrm{i}[(\mathrm{a}-\mathrm{bi})-(\mathrm{a}+\mathrm{bi})]=5$
$\Rightarrow \mathrm{i}[-2 \mathrm{bi}]=5$
$\Rightarrow \mathrm{b}=\frac{5}{2}$
So from figure it is clear that
$x=1, y=\frac{5}{2}+\frac{3}{2}=4$
$\mathrm{z}_{2}=1+4 \mathrm{i}$
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