`A(z_(1)), B(z_(2)), C(z_(3))` are the vertices of the triangle ABC (in anticlockwise order). If `angleABC=pi//4 and AB=sqrt2(BC)`, then prove that `z_(2)=z_(3) +i (z_(1)-z_(3))`

If the complex numbers z _(1) , z _(2) and z _(3) denote the vertices of an isosceles triangle, right angled at z _(1), then (z _(1) - z _(2)) ^(2) + (z _(1) - z _(3)) ^(2) is equal to A)0 B) (z _(2) + z _(3)) ^(2) C)2 D)3

If |z_(1) + z_(2)|=|z_(1)-z_(2)| , then the difference in the amplitudes of z_(1) and z_(2) is

If Im ((2z+1)/(iz+1))=3 , then locus of z is

If |z_(1)|= |z_(2)|= |z_(3)|=1 and z_(1) , z_(2), z_(3) are represented by the vertices of an equilateral triangle then which of the following is true?

If the complex numbers z_1 , z_2, z_3 and z_4 denote the vertices of a square taken in order. If z_1 = 3+4i and z_3 = 5+ 6i , then the other two vertices z_2 and z_4 are respectively a) 5+4i, 5+6i b) 5+4i, 3+6i c) 5+6i, 3+5i d) 3+6i, 5+3i

If z=1+i , then the argument of z^(2)e^(z-i) is

If arg (bar(z)_(1))= "arg" (z_(2)), (z ne 0) then

For any two complex numbers z_1 and z_2 prove that Re(z_1z_2) = Re(z_1)Re(z_2) - Im(z_1)Im(z_2)