Locate the region in the Argand plane determined by `z^2+ z ^2+2|z^2|<(8i( z -z))` .

If (log)_(sqrt(3))((|z|^2-|z|+1)/(2+|z|))>2, then locate the region in the Argand plane which represents zdot

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

Show that the area of the triangle on the Argand diagram formed by the complex number z ,i za n dz+i z is 1/2|z|^2

If z_1, z_2, z_3, z_4 are the affixes of four point in the Argand plane, z is the affix of a point such that |z-z_1|=|z-z_2|=|z-z_3|=|z-z_4| , then z_1, z_2, z_3, z_4 are

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.

P(z) be a variable point in the Argand plane such that |z| =minimum {|z−1, |z+1|} , then z+barz will be equal to a. -1 or 1 b. 1 but not equal to-1 c. -1 but not equal to 1 d. none of these

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|))

z_1a n dz_2 are two distinct points in an Argand plane. If a|z_1|=b|z_2|(w h e r ea ,b in R), then the point (a z_1//b z_2)+(b z_2//a z_1) is a point on the (a)line segment [-2, 2] of the real axis (b)line segment [-2, 2] of the imaginary axis (c)unit circle |z|=1 (d)the line with a rgz=tan^(-1)2

Let z_1, z_2a n dz_3 represent the vertices A ,B ,a n dC of the triangle A B C , respectively, in the Argand plane, such that |z_1|=|z_2|=|z_3|=5. Prove that z_1sin2A+z_2sin2B+z_3sin2C=0.

If z_1, z_2 and z_3 , are the vertices of an equilateral triangle ABC such that |z_1 -i = |z_2 -i| = |z_3 -i| .then |z_1 +z_2+ z_3| equals: