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Q.
If $z_1, z_2$ and $z_3, z_4$ are two pairs of conjugate complex numbers, then $arg. ( \frac {z_1} {z_4})$+ arg $( \frac {z_2} {z_3})$ equals
Complex Numbers and Quadratic Equations
Solution:
We have $z_{2}=\overline{z}_{1}$ and $z_{4}=\overline{z}_{3}$
$\therefore z_{1}z_{2}=z_{1}z_{1}=\left|z_{1}\right|^{2}$
$z_{3}z_{4}=z_{3}\overline{z}_{3}=\left|z_{3}\right|^{2}$
Now $arg. \left(\frac{z_{1}}{z_{4}}\right)+arg\left(\frac{z_{2}}{z_{3}}\right)=arg.\left(\frac{z_{1}z_{2}}{z_{4}z_{3}}\right)$
$=arg.\left(\frac{\left|z_{1}\right|^{2}}{\left|z_{3}\right|^{2}}\right)=arg. \left(\frac{\left|z_{1}\right|^{2}}{\left|z_{3}\right|}\right)=0$
[ $\because$ argument of $a +ve$ real number is zero]