Determine the locus of `z, z ne 21i` such that Re `((z-4)/(z-2i))=0`

Explanation

Let `z=x+iy`
then `(z-4)/(z-2i)=(x+iy-4)/(x+iy-2i)=((x-4)+iy)/(x+i(y-2))`
`=([(x-4)+iy][x-i(y-2)])/([x+i(y-2)][x-i(y-2)])`
`((x^(2)-4x+y^(2)-2y)+i(2+4y-8))/(x^(2)+(y-2)^(2))`
`:.` real part of `(z-4)/(z-4i)=(x^(2)-4x+y^(2)-2y)/(x^(2)+(y-2)^(2))`
`x^(2)-4x+y^(2)-2u=0` if and only if
`(x-2)^(2)(y-1)^(2)=5`
`z ne 2i and ("real part of" (z-4)/(z-2i))=0`
`harr (x,y) ne (0,2) and (x-2)^(2)+(y-1)^(2)=5`.
Hence hte locus of the given point representing the complex number is the circle with (2,1) as centre and `sqrt(5)`. units as radisu, excluding the point (0,2)