If the roots of `(z-1)^n=i(z+1)^n` are plotted in an Argand plane, then prove that they are collinear.

If z=omega,omega^2w h e r eomega is a non-real complex cube root of unity, are two vertices of an equilateral triangle in the Argand plane, then the third vertex z may be represented by (a) z=1 b. z=0 c. z=-2 d. z=-1

If x ,1,a n dz are in A.P. and x ,2,a n dz are in G.P., then prove that x ,a n d4,z are in H.P.

Consider an ellipse having its foci at A(z_1)a n dB(z_2) in the Argand plane. If the eccentricity of the ellipse be e an it is known that origin is an interior point of the ellipse, then prove that e in (0,(|z_1-z_2|)/(|z_1|+|z_2|))

If n >1 , show that the roots of the equation z^n=(z+1)^n are collinear.

If the imaginary part of (2z + 1)/(iz + 1) is -2 , then show that the locus of the point representing z in the argand plane is straight line.

Let vertices of an acute-angled triangle are A(z_1),B(z_2),a n dC(z_3)dot If the origin O is he orthocentre of the triangle, then prove that z_1( bar z )_2+( bar z )_1z_2=z_2( bar z )_3+( bar z )_2z_3=z_3( bar z )_1+( bar z )_(3)z_1

Z_1!=Z_2 are two points in an Argand plane. If a|Z_1|=b|Z_2|, then prove that (a Z_1-b Z_2)/(a Z_1+b Z_2) is purely imaginary.

Find the locus of point z if z ,i ,a n di z , are collinear.

z_1a n dz_2 are the roots of 3z^2+3z+b=0. if O(0),(z_1),(z_2) form an equilateral triangle, then find the value of bdot

If x=Sigma_(n=0)^(oo) a^n,y=Sigma_(n=0)^(oo) b^n,z=Sigma_(n=0)^(oo) c^n where a, b,and c are in A.P and |a|lt 1 ,|b|lt 1 and |c|1 then prove that x,y and z are in H.P