Question

If $\log _ { \sqrt { 3 } }$ $\left( \frac{|z|^{2} - |z| + 1}{2 + |z|} \right)$< 2, then the locus of z is –

• A. |z| = 5
• B. |z| < 5
• C. |z| > 5
• D. None of these

Answer 1

Answer: (B)

|z| < 5

Answer 2

Sol. We have$\log _ { \sqrt { 3 } }$ $\left( \frac{|z|^{2} - |z| + 1}{2 + |z|} \right)$ < 2 … (1)

Let $\log _ { \sqrt { 3 } }$ $\left( \frac{|z|^{2} - |z| + 1}{2 + |z|} \right)$ = k … (2)

\ $(\sqrt{3})^{k}$= $\frac{|z|^{2} - |z| + 1}{2 + |z|}$ … (3)

(2) Ž k < 2 Ž $(\sqrt{3})^{k}$< ($\sqrt{3}$)²

(3) Ž $\frac{|z|^{2} - |z| + 1}{2 + |z|}$ < 3

Ž |z|² – |z| + 1 < 6 + 3 |z|

Ž |z|² – 4 |z| – 5 < 0

Ž (|z| – 5) (|z| + 1) < 0

Ž |z| – 5 < 0, |z| + 1 > 0

Ž |z| < 5, |z| > –1 Ž |z| < 5 .