Let `z_(1)z_(2)andz_(3)` be three complex
numbers and `a,b,cin R,` such that `a+b+c=0andaz_(1)+bz_(2)+cz_(3)=0` then show that `z_(1)z_(2)and z_(3)` are
collinear.

bb"statement-1" " Let " z_(1),z_(2) " and " z_(3) be htree complex numbers, such that abs(3z_(1)+1)=abs(3z_(2)+1)=abs(3z_(3)+1) " and " 1+z_(1)+z_(2)+z_(3)=0, " then " z_(1),z_(2),z_(3) will represent vertices of an equilateral triangle on the complex plane. bb"statement-2" z_(1),z_(2),z_(3) represent vertices of an triangle, if z_(1)^(2)+z_(2)^(2)+z_(3)^(2)+z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)=0

If the complex number z_1 and z_2 be such that arg(z_1)-arg(z_2)=0 , then show that |z_1-z_2|=|z_1|-|z_2| .

Let z_(1),z_(2) " and " z_(3) be three complex numbers in AP. bb"Statement-1" Points representing z_(1),z_(2) " and " z_(3) are collinear bb"Statement-2" Three numbers a,b and c are in AP, if b-a=c-b

If iz^3+z^2-z+i = 0 , then show that |z|=1.

Let z_(1),z_(2) and z_(3) be three non-zero complex numbers and z_(1) ne z_(2) . If |{:(abs(z_(1)),abs(z_(2)),abs(z_(3))),(abs(z_(2)),abs(z_(3)),abs(z_(1))),(abs(z_(3)),abs(z_(1)),abs(z_(2))):}|=0 , prove that (i) z_(1),z_(2),z_(3) lie on a circle with the centre at origin. (ii) arg(z_(3)/z_(2))=arg((z_(3)-z_(1))/(z_(2)-z_(1)))^(2) .

If z_(1) , z_(2) , z_(3) are any three complex numbers on Argand plane, then z_(1)(Im(barz_(2)z_(3)))+z_(2)(Imbarz_(3)z_(1)))+z_(3)(Imbarz_(1)z_(2))) is equal to

Consider z_(1)andz_(2) are two complex numbers such that |z_(1)+z_(2)|=|z_(1)|+|z_(2)| Statement -1 amp (z_(1))-amp(z_(2))=0 Statement -2 The complex numbers z_(1) and z_(2) are collinear. Check for the above statements.

If z_(1),z_(2) and z_(3), z_(4) are two pairs of conjugate complex numbers, the find the value of arg(z_(1)/z_(4)) + arg(z_(2)//z_(3)) .

If two complex numbers z_1,z_2 are such that |z_1|= |z_2| , is it then necessary that z_1 = z_2 ?

The number of complex numbers z such that |z-1|=|z+1|=|z-i| is