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4-1.Complex numbers
medium
જો $\frac{{2{z_1}}}{{3{z_2}}}$ એ શુદ્ધ કાલ્પનિક સંખ્યા હોય તો $\left| {\frac{{{z_1} - {z_2}}}{{{z_1} + {z_2}}}} \right|$ = . . .
A
$1.5$
B
$1$
C
$2/3$
D
$4/9$
Solution
(b) As given, let $\frac{{2{z_1}}}{{3{z_2}}} = iy$ or $\frac{{{z_1}}}{{{z_2}}} = \frac{3}{2}iy$, so that
$\left| {\frac{{{z_1} – {z_2}}}{{{z_1} + {z_2}}}} \right| = \left| {\frac{{\frac{{{z_1}}}{{{z_2}}} – 1}}{{\frac{{{z_1}}}{{{z_2}}} + 1}}} \right| = \left| {\frac{{\frac{3}{2}iy – 1}}{{\frac{3}{2}iy + 1}}} \right| = \left| {\frac{{1 – \frac{3}{2}iy}}{{1 + \frac{3}{2}iy}}} \right| = 1$
$\left\{ {\because \,\,\,\,|z|\, = \,|\overline z |} \right\}$
Standard 11
Mathematics
