Mathematics - Complex Number Question with Solution | TestHub

Let$z_1=x_1+i{y}_1,z_2=x_2+i{y}_2$
$z_1-z_2=\left(x_1-x_2\right)+i\left(y_1-y_2\right)$
$\because \arg \left(z_1-z_2\right)=\frac{\pi }{4}\Rightarrow {\tan }^{-1}\left(\frac{y_1-y_2}{x_1-x_2}\right)=\frac{\pi }{4}$
$\Rightarrow y_1-y_2=x_1-x_2----\left(1\right)$
$\left|z_1-3\right|=Re\left(z_1\right)\Rightarrow {\left(x_1-3\right)}^2+y_1^2=x_1^2$
$\left|z_2-3\right|=Re\left(z_2\right)\Rightarrow {\left(x_2-3\right)}^2+y_2^2=x_2^2$
$\Rightarrow {\left(x_1-3\right)}^2-{\left(x_2-3\right)}^2+\left(y_1^2-y_2^2\right)=x_1^2-x_2^2$
$\Rightarrow \left(x_1-x_2\right)\left(x_1+x_2-6\right)+\left(y_1-y_2\right)\left(y_1+y_2\right)=\left(x_1+x_2\right)\left(x_1-x_2\right)$
$\Rightarrow x_1+x_2-6+y_1+y_2=x_1+x_2$
$\Rightarrow y_1+y_2=6$