Question

If for the complex numbers z₁& z₂

$|1–{\overline{z}}_{1}z_{2}|^{2}$– | z₁ – z₂|² = k (1 – | z |²) (1 – |z₂|²), then k equals –

• A. 1
• B. –1
• C. 2
• D. 4

Answer 1

Answer: (A)

1

Answer 2

Sol. $\left( \overline{1–\overline{z_{1}}}z_{2} \right)$– (z₁ – z₂)$\left( \overline{z_{1}–z_{2}} \right)$

= $\left( 1–\overline{z_{1}z_{2}} \right)$ – (z₁ – z₂)$\left( \overline{z_{1}–z_{2}} \right)$

= $\left( 1–z_{1}{\overline{z}}_{2} \right)$ – (z₁ – z₂)$\left( {\overline{z}}_{1}–{\overline{z}}_{2} \right)$

=1– ${\overline{z}}_{1}z_{2}$–$z_{1}{\overline{z}}_{2}$+ ${\overline{z}}_{1}z_{2}$ – $z_{1}{\overline{z}}_{1}$+ $z_{2}{\overline{z}}_{1}$+ $z_{1}{\overline{z}}_{2}$– $z_{2}{\overline{z}}_{2}$

= 1+| z₁ |² | z₂|² – | z₁ |² – $|z_{2}^{2}|$= (1 – | z₁ |²) (1 – | z |²)

\ k = 1