If \(\operatorname{Re}\left(\frac{z-1}{2 z+i}\right)=1\), where z = x + iy, then the point (x, y) lies on a

Explanation -

z = x + iy

\(\frac{x+i y-1}{2 x+2 i y+i}=\frac{(x-1)+i y}{2 x+i(2 y+1)}\left(\frac{2 x-i(2 y+1)}{2 x-i(2 y+1)}\right)\)

\(\frac{2 x(x-1)+y(2 y+1)}{4 x^2+(2 y+1)^2}=1\)

2x2 + 2y2 – 2x + y = 4x2 + 4y2 + 4y + 1

x2 + y2 + x + (3/2)y + (1/2) = 0

Circle’s centre will be (-1/2, -3/4)

Radius = \(\sqrt{[(1/4) + (9/16) – (1/2)]} = \sqrt 5/4\)

Diameter = √5/2

Hence Option (3) is correct.