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Solve the equation |z| = z + 1 + 2i

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Best answer

Given: 

⇒ |z|=z+1+2i 

Let us assume z=x+iy 

⇒ |x+iy|=x+iy+1+2i

\(\sqrt{x^2+y^2}\) = (x+1)+i(y+2)

Equating Real and Imaginary parts on both sides we get 

⇒ y+2=0 

⇒ y=-2 ......... (1)

⇒ \(\sqrt{x^2+y^2}\) (x+1)

⇒ x2+(-2)2=(x+1)2 

⇒ x2+4=x2+2x+1 

⇒ 2x=3

⇒ x = \(\frac{3}{2}\) 

∴ z = \(\frac{3}{2}\) - 2i

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