Let z _ 1 and z _ 2 be two complex numbers such that arg( z _ 1 - z...

Let ${z}_{1} = {x}_{1} + i{y}_{1}, {z}_{2} = {x}_{2} + i{y}_{2}$
${z}_{1} - {z}_{2} = \left({x}_{1} - {x}_{2}\right) + i\left({y}_{1} - {y}_{2}\right)<br />$∵ arg \left({z}_{1} - {z}_{2}\right)=\dfrac{\pi}{4}$⇒$\tan^{- 1}\left(\dfrac{{y}_{1} - {y}_{2}}{{x}_{1} - {x}_{2}}\right) = \dfrac{\pi}{4}$<br />$⇒ {y}_{1} - {y}_{2} = {x}_{1} - {x}_{2} - - - - ( 1 )$<br />\left|{z}_{1} - 3\right|$ = Re \left({z}_{1}\right)$⇒ {\left({x}_{1} - 3\right)}$^{2} + {y}_{1}^{2} = {x}_{1}^{2}$<br />\left|{z}_{2} - 3\right| = Re \left({z}_{2}\right) ⇒ {\left({x}_{2} - 3\right)}^{2} + {y}_{2}^{2} = {x}_{2}^{2}$
$⇒ {\left({x}_{1} - 3\right)}^{2} - {\left({x}_{2} - 3\right)}^{2} + \left({y}_{1}^{2} - {y}_{2}^{2}\right) = {x}_{1}^{2} - {x}_{2}^{2}$
$⇒ \left({x}_{1} - {x}_{2}\right) \left({x}_{1} + {x}_{2} - 6\right) + \left({y}_{1} - {y}_{2}\right) \left({y}_{1} + {y}_{2}\right) = \left({x}_{1} + {x}_{2}\right) \left({x}_{1} - {x}_{2}\right)$
$⇒ {x}_{1} + {x}_{2} - 6 + {y}_{1} + {y}_{2} = {x}_{1} + {x}_{2}$
$⇒ {y}_{1} + {y}_{2} = 6$