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4-1.Complex numbers
hard
If ${z_1}$ and ${z_2}$ are two complex numbers satisfying the equation $\left| \frac{z_1 +z_2}{z_1 - z_2} \right|=1$, then $\frac{{{z_1}}}{{{z_2}}}$ is a number which is
A
Positive real
B
Negative real
C
Zero or purely imaginary
D
None of these
Solution
(c)Given $\left| {\frac{{{z_1} + {z_2}}}{{{z_1} – {z_2}}}} \right| = 1$==> $\frac{{{z_1} + {z_2}}}{{{z_1} – {z_2}}} = \cos \theta + i\sin \theta $(say)
==> $\frac{{{z_1}}}{{{z_2}}} = \frac{{1 + \cos \theta + i\sin \theta }}{{ – 1 + \cos \theta + i\sin \theta }} = – i\cot \frac{\theta }{2}$
which is zero, if $\theta = n\pi (n \in I),$ and is otherwise purely imaginary.
Standard 11
Mathematics
