As per the inquiry,
We have,
![\[\left| z \right|\text{ }=\text{ }z\text{ }+\text{ }1\text{ }+\text{ }2i\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b604bc6bde21a644277a719c9e782e3b_l3.png "Rendered by QuickLaTeX.com")
Subbing z = x + iy, we get,
![\[\begin{array}{*{35}{l}} \Rightarrow \left| x\text{ }+\text{ }iy \right|\text{ }=\text{ }x\text{ }+\text{ }iy\text{ }+\text{ }1\text{ }+\text{ }2i \\ ~ \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-395512cb62c561e928f1b9f7af936790_l3.png "Rendered by QuickLaTeX.com")
We realize that,
![\[\left| z \right|\text{ }=\text{ }\surd \left( x^2\text{ }+\text{ }y^2 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d1930a643f4a75a60accabc48cfc6d29_l3.png "Rendered by QuickLaTeX.com")
![\[\surd \left( x^2\text{ }+\text{ }y^2 \right)\text{ }=\text{ }\left( x\text{ }+\text{ }1 \right)\text{ }+\text{ }i\left( y\text{ }+\text{ }2 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5f99eb0ecb5248c00a0179fd04c4fd30_l3.png "Rendered by QuickLaTeX.com")
Looking at genuine and nonexistent parts,
We get,
![\[\surd \left( x^2\text{ }+\text{ }y^2 \right)\text{ }=\text{ }\left( x\text{ }+\text{ }1 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4841aca4dc20131b5e248c4ffda082fe_l3.png "Rendered by QuickLaTeX.com")
Also, 0 = y + 2
⇒ y = – 2
Subbing the worth of y in
![\[\surd \left( x^2\text{ }+\text{ }y^2 \right)\text{ }=\text{ }\left( x\text{ }+\text{ }1 \right),\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-cd8f68f8abd5a4cfdd0c9bb42056a4e7_l3.png "Rendered by QuickLaTeX.com")
We get,
![\[\Rightarrow x^2\text{ }+\text{ }\left( -\text{ }2 \right)^2\text{ }=\text{ }\left( x\text{ }+\text{ }1 \right)^2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b7d2e3554c4c86cc1798fbbd43f5fca5_l3.png "Rendered by QuickLaTeX.com")
![\[\Rightarrow x^2\text{ }+\text{ }4\text{ }=\text{ }x^2\text{ }+\text{ }2x\text{ }+\text{ }1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2cdfe05c58a9aadbdca4e29c87993ec0_l3.png "Rendered by QuickLaTeX.com")
Consequently, x = 3/2
Consequently, z = x + iy
= 3/2 – 2i