If the real part of `(barz +2)/(barz-1)` is 4, then show that the locus of the point representing z in the complex plane is a circle.

Explanation

Let z = x +`i`y
Now, `(barz + 2)/(barz -1) =(x - `i`y +2)/(x-`i`y-1)`
`=([(x+2)-i y][(x -1)+ i y])/([(x -1) - i y ][(x -1)+ i y])`
=`((x -1)(x+2)-iy(x -1)+i y(x + 2) + y^(2))/((x -1)^(2) +y^(2))`
=`((x -1)(x+2)+y^(2)+i[(x + 2)y-(x - 1) y])/((x -1)^(2) +y^(2)) " "[:. -i^(2) = 1]`
Taking real part, `((x-1)(x +2)+ y^(2))/((x - 1)^(2) +Y^(2)) = 4`
`rArr x^(2) - x + 2x - 2 + y^(2) = 4 (x^(2) - 2x + 1 + Y^(2))`
`rArr 3x^(2) + 3y^(2) - 9x + 6 = 0. which reperesents a circle`.
Hence, z lines on the circle.