Prove that `|Z-Z_1|^2+|Z-Z_2|^2=a` will represent a real circle [with center `(|Z_1+Z_2|^//2+)` ] on the Argand plane if `2ageq|Z_1-Z_1|^2`

Prove that |z_1+z_2|^2=|z_1|^2+|z_2|^2, ifz_1//z_2 is purely imaginary.

Let z be not a real number such that (1+z+z^2)//(1-z+z^2) in R , then prove tha |z|=1.

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_1^2+z_2^2+2z_1.z_2.costheta= 0 prove that the points represented by z_1, z_2 , and the origin form an isosceles triangle.

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

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

If z_0 is the circumcenter of an equilateral triangle with vertices z_1, z_2, z_3 then z_1^2+z_2^2+z_3^2 is equal to

For all complex numbers z_1,z_2 satisfying |z_1|=12 and |z_2-3-4i|=5 , find the minimum value of |z_1-z_2|

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.

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