Given z1=1+2i and let z2=x+iy .
(i) We have ∣z1+z2∣=∣z1∣+∣z2∣
⟹∣(1+x)+i(2+y)∣=∣1+2i∣+∣x+iy∣
⟹(1+x)2+(2+y)2=5+x2+y2
By squaring on both sides, we getx2+y2+2x+4y+5=5+x2+y2+25x2+5y2
⟹x+2y=5x2+5y2
Again by squaring on both side, we get x2+4y2+4xy=5x2+5y2
⟹4x2+y2−4xy=0
⟹(2x−y)2=0⟹2x−y=0⟹y=2x .
Hence, z2=x(1+2i) .
(ii) We have ∣z1+z2∣=∣z1∣−∣z2∣
⟹∣(1+x)+i(2+y)∣=∣1+2i∣−∣x+iy∣
⟹(1+x)2+(2+y)2=5−x2+y2
By squaring on both sides, we get
x2+y2+2x+4y+5=5+x2+y2−25x2+5y2
⟹x+2y=−5x2+5y2
Again by squaring on both side, we get x2+4y2+4xy=5x2+5y2
⟹4x2+y2−4xy=0
⟹(2x−y)2=0⟹2x−y=0⟹y=2x .
Hence, z2=x(1+2i) .
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