If the imaginary part of `(2z+1)/(iz+1)` is -2, then the locus of the point representing z in the complex plane is :

If |z|=2, the points representing the complex numbers -1+5z will lie on

If z ne 1 and (z^(2))/(z-1) is real, then the point represented by the complex number z lies

Represent the following complex numbers in the complex plane : 1 + 2i.

The centre of circle represented by |z + 1| = 2 |z - 1| in the complex plane is

locus of the point z satisfying the equation |iz-1|+|z-i|=2 is

Write the complex numbers that represent the following points in the plane : (1, - 2).

Write the complex numbers that represent the following points in the plane : (-1/2,-1/2) .

Let z_1 \and\ z_2 be the roots of the equation z^2+p z+q=0, where the coefficients p \and \q may be complex numbers. Let A \and\ B represent z_1 \and\ z_2 in the complex plane, respectively. If /_A O B=theta!=0 \and \O A=O B ,\where\ O is the origin, prove that p^2=4q"cos"^2(theta//2)dot

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

A,B and C are the points respectively the complex numbers z_(1),z_(2) and z_(3) respectivley, on the complex plane and the circumcentre of /_\ABC lies at the origin. If the altitude of the triangle through the vertex. A meets the circumcircle again at P, prove that P represents the complex number (-(z_(2)z_(3))/(z_(1))) .