![\[\left| z\text{ }+\text{ }1 \right|\text{ }=\text{ }z\text{ }+\text{ }2\text{ }\left( 1\text{ }+\text{ }i \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-058b280688dcf790bbd8b9fa1f840985_l3.png "Rendered by QuickLaTeX.com")
Substituting z = x + iy, we get,
![\[\Rightarrow ~\left| x\text{ }+\text{ }iy\text{ }+\text{ }1 \right|\text{ }=\text{ }x\text{ }+\text{ }iy\text{ }+\text{ }2\text{ }\left( 1\text{ }+\text{ }i \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-349649c0fe913691cc892cb0b6faa248_l3.png "Rendered by QuickLaTeX.com")
We know,
![\[\left| z \right|\text{ }=\text{ }\surd ({{x}^{2}}~+\text{ }{{y}^{2}})\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d433c0c2df3dd75cf2343e4c912fdb87_l3.png "Rendered by QuickLaTeX.com")
![\[\surd ({{\left( x\text{ }+\text{ }1 \right)}^{2}}~+\text{ }{{y}^{2}})\text{ }=\text{ }\left( x\text{ }+\text{ }2 \right)\text{ }+\text{ }i\left( y\text{ }+\text{ }1 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-675457dbd392da3916e0fb9455b299e7_l3.png "Rendered by QuickLaTeX.com")
Comparing real and imaginary parts,
![\[\Rightarrow \surd ({{\left( x\text{ }+\text{ }1 \right)}^{2}}~+\text{ }{{y}^{2}})\text{ }=\text{ }x\text{ }+\text{ }2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9ec62f6bb5ab7754ea2fe0fcfd53a34b_l3.png "Rendered by QuickLaTeX.com")
And 0 = y + 2
⇒ y = -2
Substituting the value of y in
![\[\surd ({{\left( x\text{ }+\text{ }1 \right)}^{2}}~+\text{ }{{y}^{2}})\text{ }=\text{ }x\text{ }+\text{ }2,\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0dc3c2122d064fce72f4071620cc26c2_l3.png "Rendered by QuickLaTeX.com")
![\[\Rightarrow ~{{\left( x\text{ }+\text{ }1 \right)}^{2}}~+\text{ }{{\left( -2 \right)}^{2}}~=\text{ }{{\left( x\text{ }+\text{ }2 \right)}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0b5284dbf3a7ae9081c0b553322421d5_l3.png "Rendered by QuickLaTeX.com")
![\[\Rightarrow ~{{x}^{2}}~+\text{ }2x\text{ }+\text{ }1\text{ }+\text{ }4\text{ }=\text{ }{{x}^{2}}~+\text{ }4x\text{ }+\text{ }4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d95b5f9fb3177f1313837c95eaed29e4_l3.png "Rendered by QuickLaTeX.com")
⇒ 2x = 1
Hence, x = ½
Hence, z = x + iy
= ½ – 2i
Substituting z = x + iy, we get,
We know,
Comparing real and imaginary parts,
And 0 = y + 2
⇒ y = -2
Substituting the value of y in
⇒ 2x = 1
Hence, x = ½
Hence, z = x + iy
= ½ – 2i