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  4. If z1, z2 and z3, z4 are 2 pairs of complex conjugate numbers, then arg ((z1/z4))+ arg ((z2/z3)) equals :

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Q. If $z_1, \, z_2$ and $z_3, \, z_4$ are 2 pairs of complex conjugate numbers, then arg $\left(\frac{z_{1}}{z_{4}}\right)+ arg \left(\frac{z_{2}}{z_{3}}\right)$ equals :

JEE MainJEE Main 2014Complex Numbers and Quadratic Equations

Solution:

$Z_{2} = \bar{Z_{1}}\,\&\, Z_{4} = \bar{Z_{3}}$
$arg \left(\frac{Z_{1}}{Z_{4}}\right) + arg \left(\frac{Z_{2}}{Z_{3}}\right)$
$= arg \,Z_{1} - arg\, Z_{4} + arg \,Z_{2} - arg \,Z_{3}$
$ = arg\, Z_{1} - arg \,\bar{Z}_{3} + arg \,\bar{Z}_{1} - arg \,Z_{3}$
$= arg \,Z_{1} + arg\, Z_{3} - arg\, Z_{1} - arg \,Z_{3} = 0$