Question

If Im$\left( \frac{2z + 1}{iz + 1} \right) = - 2$, then the locus of the point representing z in the complex plane is

• A. Circle
• B. A straight line
• C. A parabola
• D. None of these

Answer 1

Answer: (B)

A straight line

Answer 2

Sol. Im$\left( \frac{2z + 1}{iz + 1} \right) = - 2$

⇒ $\frac{1}{2i}\left( \frac{2z + 1}{iz + 1} - \frac{2\bar{z} + 1}{- i\bar{z} + 1} \right)$ = – 2

⇒ (1+ 2z) (1 – i$\bar{z}$) – (2$\bar{z}$ + 1) (iz + 1) = – 4i (iz + 1)

(1 – i$\bar{z}$)

⇒ 1 – i$\bar{z}$ + 2z – 2 iz$\bar{z}$ – 2iz$\bar{z}$– 2$\bar{z}$ – iz – 1

= – 4i (iz + z$\bar{z}$ + 1 – i$\bar{z}$)

⇒ 2(z – $\bar{z}$) – i(z +$\bar{z}$) – 4iz$\bar{z}$

= – 4i$\left\lbrack z\bar{z} + 1 + i\left( z - \bar{z} \right) \right\rbrack$.…. (1)

⇒ Let z = x + iy, x, y, ∈ R.

Then (1) yields x + 2y = 2 which gives a straight line