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4-1.Complex numbers
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Let ${z_1}$ be a complex number with $|{z_1}| = 1$ and ${z_2}$be any complex number, then $\left| {\frac{{{z_1} - {z_2}}}{{1 - {z_1}{{\bar z}_2}}}} \right| = $
A
$0$
B
$1$
C
$-1$
D
$2$
Solution
(b) We have $|{z_1}|\; = 1$ and ${z_2}$be any complex number.
$ \Rightarrow \left| {\;\frac{{{z_1} – {z_2}}}{{1 – {z_1}{{\bar z}_2}}}} \right|\; = \frac{{|{z_1} – {z_2}|}}{{\left| {\;1 – \frac{{{{\bar z}_2}}}{{{{\bar z}_1}}}\;} \right|}}$;$\because \;{z_1}{\bar z_1} = \;|{z_1}{|^2}$
$ = \frac{{|{z_1} – {z_2}|}}{{|{{\bar z}_1} – {{\bar z}_2}|}}|{\bar z_1}|$; Given that $\;|{\bar z_1}|\; = 1$
$ = \frac{{|{z_1} – {z_2}|}}{{|\overline {{z_1} – {z_2}} |}} = \frac{{|{z_1} – {z_2}|}}{{|{z_1} – {z_2}|}} = 1$.
Standard 11
Mathematics
