Question

if z = x + iy, `w = (2 -iz)/(2z - i)` and |w| = 1. Find the locus of z and illustrate it in the Argand Plane.

Answer 1

z =x + iy

`w = (2 - iz)/(2z - i)`

`= (2 - i(x + iy))/(2(x + iy) - i)`

`= (2 - ix - i^2y)/(2x + 2iy - i)`

`w = ((2 + y) - ix)/(2x +(2y - 1)i)`

`|w| = |((2+y) - ix)/(2x + (2y - 1)i)|`

`1 = sqrt((2+y)^2 + x^2)/sqrt((2x)^2 + (2y - 1)^2)`

`4x^2 + 4y^2 - 4y + 1 = 4 +4y + y^2   + x^2`

`3x^2 + 3y^2 - 8y - 3 = 0`

`x^2 + y^2  - 8/3 y - 1 = 0`

∴ Locus is a circle

with centre `(0, 4/3)`

Radius = `sqrt(16/9 + 1) = 5/3`