For any three complex numbers `Z_1,Z_2,Z_3` if `Delta=|(1,Z_1,overlineZ_1),(1,Z_2,overlineZ_2),(1,Z_3,overlineZ_3)|,`

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

In complex plane z_(1), z_(2) and z_(3) be three collinear complex numbers, then the value of |(z_(1),barz_(1),1),(z_(2),barz_(2),1),(z_(3), barz_(3),1)| is -

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 .

If z_1, z_2, z_3 are three complex numbers such that 5z_1-13 z_2+8z_3=0, then prove that [(z_1,(bar z )_1, 1),(z_2,(bar z )_2 ,1),(z_3,(bar z )_3 ,1)]=0

If z_1, z_2, z_3 are three complex numbers such that 5z_1-13 z_2+8z_3=0, then prove that [(z_1,(bar z )_1, 1),(z_2,(bar z )_2 ,1),(z_3,(bar z )_3 ,1)]=0

If z_1, z_2, z_3 are three complex numbers such that 5z_1-13 z_2+8z_3=0, then prove that [(z_1,(bar z )_1, 1),(z_2,(bar z )_2 ,1),(z_3,(bar z )_3 ,1)]=0

Let z_1=2-i, z_2=-2+i . Find Im (1/(z_1overlinez_1))

Let z and w are two non zero complex number such that |z|=|w|, and Arg(z)+Arg(w)=pi then (a) z=w (b) z=overlinew (c) overlinez=overlinew (d) overlinez=-overlinew