If w = z z - 1 3 i and | w |=1, i = -1 , then z lies on

Step-by-step solution

aligned & w= z z- 1 3 i & w= 3 z 3 z-i aligned Applying mod on both sides, we get aligned & |w|= 3|z| |3 z-i| & 3|z|=|3 z-i| [|w|=1] aligned Consider z = a + ib aligned & 3|a+i b|=|3 a+3 i b-i| & 3|a+i b|=|3 a+(3 b-1) i| & 3 ( a^2+b^2 )= ( 9 a^2+(3 b-1)^2 ) & 9 a^2+9 b^2=9 a^2+9 b^2-6 b+1 & 6 b-1=0 aligned The above equation represents a straight line.