For any two complex numbers `z_(1)` and `z_(2)`, prove that `Re (z_(1)z_(2)) = Re z_(1) Re z_(2)- Imz_(1) Imz_(2)`

In the complex plane, the vertices of an equlitateral triangle are represented by the complex numbers z_(1),z_(2) and z_(3) prove that, (1)/(z_(1)-z_(2))+(1)/(z_(2)-z_(3))+(1)/(z_(3)-z_(1))=0

The vertices A,B,C of an isosceles right angled triangle with right angle at C are represented by the complex numbers z_(1) z_(2) and z_(3) respectively. Show that (z_(1)-z_(2))^(2)=2(z_(1)-z_(3))(z_(3)-z_(2)) .

For two unimobular complex numbers z_(1) and z_(2) , find [(bar(z)_(1),-z_(2)),(bar(z)_(2),z_(1))]^(-1) [(z_(1),z_(2)),(-bar(z)_(2),bar(z)_(1))]^(-1)

If z_(1)=-3+4i and z_(2)=12-5i, "show that", |z_(1)z_(2)|=|z_(1)||z_(2)|

If z_(1)=4-3iand z_(2)=-12+5i, "show that", |z_(1)z_(2)|=|z_(1)||z_(2)|

If z_(1)=4-3iand z_(2)=-12+5i, "show that", |z_(1)/z_(2)|=|z_(1)|/|z_(2)|

For any complex number z,prove that |Re(z)|+|I_m(z)|le\sqrt 2 |z|

If z_(1),z_(2) are two complex numbers , prove that , |z_(1)+z_(2)|le|z_(1)|+|z_(2)|

If z_(1)=-3+4i and z_(2)=12-5i, "show that", |(z_(1))/(z_(2))|=(|z_(1)|)/(|z_(2)|)

z_(1) and z_(2)( ne z_(1)) are two complex numbers such that |z_(1)|=|z_(2)| . Sho that the real part of (z_(1)+z_(2))/(z_(1)-z_(2)) is zero.