1.

Find the complex number z for which |z| = z + 1 + 2i.

Answer» Correct Answer - `((2)/(2)-2i)`
Let the required complex number be z = (x + iy). Then, |z| = z + 1 + 2i
`rArr" "|x+iy|=(x+iy)+1+2i`
`rArr" "sqrt(x^(2)+y^(2))=(x+1)+(y+2)i`
`rArr" "sqrt(x^(2)+y^(2))=(x+1) and y + 2 = 0`
`rArr" "y = -2 and sqrt(x^(2)+(-2)^(2))=(x+1)`
`rArr" "y = -2 and x^(2) + 4 = (x + 1)^(2) rArr x = (3)/(2) and y = -2`.


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