[" 18.The points "P,Q" denote the complex numbers "z_(1),z_(2)" in the argand diagram.O "],[" is the origin.If "z_(1)bar(z)_(2)+bar(z)_(1)z_(2)=0," show that "/_POQ=90^(@).]

The points P,Q denote the complex numbers, barz_(1),barz_(2) in the Argand diagram. O. is the origin. If bar(z_(1)z_(2))+ bar (z_(2)z_(1))=0 , show that anglePOQ=90^(@)

If the vertices of an equilateral triangle be represented by the complex numbers z_(1),z_(2),z_(3) on the Argand diagram, then prove that, z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1).

If z_(1) and z_(2) are two complex number such that |z_(1)|<1<|z_(2)| then prove that |(1-z_(1)bar(z)_(2))/(z_(1)-z_(2))|<1

Let Z_(1) and z_(2) be two complex numbers with z_(1)!=z_(2) if |z_(1)|=sqrt(2), then |(z_(1)-bar(z)_(2))/(2-z_(1)z_(2))| equals

For any three complex numbers z_1, z_2 and z_3 , prove that z_1 lm (bar(z_2) z_3)+ z_2 lm (bar(z_3) z_1) + z_3lm(bar(z_1) z_2)=0 .

z_(1)bar(z)_(2)+bar(z)_(1)z_(2)=2|z_(1)||z_(2)|cos(theta_(1)-theta_(2))

For any two non zero complex numbers z_(1) and z_(2) if z_(1)bar(z)_(2)+z_(2)bar(z)_(1)=0 then amp(z_(1))-amp(z_(2)) is

For any complex numbers z_(1),z_(2) and z_(3),z_(3)Im(bar(z_(2))z_(3))+z_(2)Im(bar(z_(3))z_(1))+z_(1)Im(bar(z_(1))z_(2)) is

For any three complex numbers z_(1),z_(2),z_(3) , if Delta=|{:(1,z_(1),bar(z_(1))),(1,z_(2),bar(z_(2))),(1,z_(3),bar(z_(3))):}| , then

The complex numbers z_(1), z_(2) and the origin are the vertices of an equilateral triangle in the Argand plane Prove that z_(1)^(2)+ z_(2)^(2)= z_(1).z_(2)