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4-1.Complex numbers
hard
જો $z$ અને $w$ સંકર સંખ્યા છે કે જેથી $|zw| = 1$ અને $arg(z) -arg(w) =\frac {\pi }{2},$ થાય તો .........
A
$\bar zw\,\, = \,i$
B
$z\bar w\,\, = \,\frac{{ - 1 + i}}{{\sqrt 2 }}$
C
$z\bar w\,\, = \,\frac{{1 - i}}{{\sqrt 2 }}$
D
$\bar zw\,\, = - \,i$
(JEE MAIN-2019)
Solution
$|z| \cdot|w|=1 \quad z =$ $r e^{\left(\theta+\frac{\pi}{2}\right)} \text { and } w$ $=\frac{1}{r} e^{i \theta}$
$\bar{z} \cdot w$ $=e^{-i\left(\theta+\frac{\pi}{2}\right)} \cdot e^{i \theta} $ $=e^{-i\left(\frac{\pi}{2}\right)}=-i$
$\mathrm{z} \cdot \overline{\mathrm{w}}$ $=\mathrm{e}^{i\left(\theta+\frac{\pi}{2}\right)} \cdot \mathrm{e}^{-\mathrm{i} \theta}$ $=\mathrm{e}^{\mathrm{i}\left(\frac{\pi}{2}\right)}=\mathrm{i}$
Standard 11
Mathematics
