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4-1.Complex numbers
easy
જો $|z|\, = 1,(z \ne - 1)$ અને $z = x + iy$ તો $\left( {\frac{{z - 1}}{{z + 1}}} \right)$ =. . .
A
શુદ્ધ વાસ્તવિક
B
શુદ્ધ કાલ્પનિક
C
શૂન્ય
D
અવ્યાખ્યાતીત
Solution
(b)$z = x + iy \Rightarrow |z{|^2} = {x^2} + {y^2} = 1$ …..$(i)$
Now, $\left( {\frac{{z – 1}}{{z + 1}}} \right) = \frac{{(x – 1) + iy}}{{(x + 1) + iy}} \times \frac{{(x + 1) – iy}}{{(x + 1) – iy}}$
$ = \frac{{({x^2} + {y^2} – 1) + 2iy}}{{{{(x + 1)}^2} + {y^2}}}$$ = \frac{{2iy}}{{{{(x + 1)}^2} + {y^2}}}$ [by equation $(i)$]
Hence, $\left( {\frac{{z – 1}}{{z + 1}}} \right)$is purely imaginary.
Standard 11
Mathematics
