If `arg (z^(1//2))=(1)/(2)arg(z^(2)+barzz^(1//3)),` find the value of `|z|.`

If arg (z_(1))=(17pi)/18 and arg (z_(2))=(7pi)/18, find the principal argument of z_(1)z_(2) and (z_(1)//z_(2)).

If |z_(1)| = 1, |z_(2)| =2, |z_(3)|=3, and |9z_(1)z_(2) + 4z_(1)z_(3)+ z_(2)z_(3)|= 12 , then find the value of |z_(1) + z_(2) + z_(3)| .

If z=(-2)/(1+sqrt(3) i) , then the value of arg z is

If the lines (x-1)/(-3)=(y-2)/(2k)=(z-3)/2 and (x-1)/(3k)=(y-1)/1=(z-6)/(-3) are perpendicular, find the value of k.

If Im((z-1)/(2 z+1))=-4 then the locus of z is

If |z^(2)-1|=|z|^(2)+1 , then z lies on :

If |z+(4)/(z)|=2, find the maximum and minimum values of |z|.

If |z_(1)|=|z_(2)| and arg z_(1) + arg z_(2)=0 then

For all complex numbers z_(1), z_(2) satisfying |z_(1)|=12 and |z_(2)-3-4 i|=5 , the minimum value of |z_(1)-z_(2)| is

The complex numbers z_(1), z_(2), z_(3) satisfying (z_(1)-z_(3))/(z_(2)-z_(3))=(1-i sqrt(3))/(2) are the vertices of a triangle which is