Question

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.

Answer 1

Let z = x + iy

∴ `barz` = x – iy

So`(barz + 2)/(barz - 1) = (x - iy + 2)/(x - iy - 1)`

= `((x + 2) - iy)/((x - 1) - iy)`

= `((x + 2) - iy)/((x - 1) - iy) xx ((x - 1) + iy)/((x - 1) + iy)`

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

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

= `(x^2 + y^2 + x - 2)/((x - 1)^2 + y^2) + (3y)/((x - 1)^2 + y^2)i`

Real part = 4

∴ `(x^2 + y^2 + x - 2)/((x - 1)^2 + y^2)` = 4

⇒ x² + y² + x – 2 = 4[(x – 1)² + y²]

⇒ x² + y² + x – 2 = 4[x² + 1 – 2x + y²]

⇒ x² + y² + x – 2 = 4x² + 4 – 8x + 4y²

⇒ x² – 4x² + y² – 4y² + x + 8x – 2 – 4 = 0

⇒ – 3x² – 3y² + 9x – 6 = 0

⇒ x² + y² – 3x + 2 = 0

Which represents a circle.

Hence, z lies on a circle.