Let `a ,b ,a n dc` be any three nonzero complex number. If `|z|=1a n d' z '` satisfies the equation `a z^2+b z+c=0,` prove that `a ( bara) =c (barc) a n d|a||b|=sqrt(a c( bar b )^2)`

If a ,b ,c are nonzero complex numbers of equal moduli and satisfy a z^2+b z+c=0, hen prove that (sqrt(5)-1)//2lt=|z|lt=(sqrt(5)+1)//2.

Find the number of complex numbers which satisfies both the equations |z-1-i|=sqrt(2)a n d|z+1+i|=2.

Let za n dw be two nonzero complex numbers such that |z|=|w|a n d arg(z)+a r g(w)=pidot Then prove that z=- bar w dot

Modulus of nonzero complex number z satisfying barz +z=0 and |z|^2-4z i=z^2 is _________.

The number of complex numbers z such that |z|=1 and |z/ barz + barz/z|=1 is arg(z) in [0,2pi)) then a. 4 b. 6 c. 8 d. more than 8

Let a , b , c be distinct complex numbers with |a|=|b|=|c|=1 and z_(1) , z_(2) be the roots of the equation az^(2)+bz+c=0 with |z_(1)|=1 . Let P and Q represent the complex numbers z_(1) and z_(2) in the Argand plane with /_POQ=theta , o^(@) lt 180^(@) (where O being the origin).Then

Let z_1, z_2, z_3 be the three nonzero complex numbers such that z_2!=1,a=|z_1|,b=|z_2|a n d c=|z_3|dot Let |(a, b, c), (b, c, a), (c,a,b) |=0 a r g(z_3)/(z_2) equal to (a) arg((z_3-z_1)/(z_2-z_1))^2 (b) orthocentre of triangle formed by z_1, z_2, z_3, i sz_1+z_2+z_3 (c)if triangle formed by z_1, z_2, z_3 is equilateral, then its area is (3sqrt(3))/2|z_1|^2 (d) if triangle formed by z_1, z_2, z_3 is equilateral, then z_1+z_2+z_3=0

a , b , c are three complex numbers on the unit circle |z|=1 , such that abc=a+b+c . Then |ab+bc+ca| is equal to

Let z_1, z_2, z_3 be three complex numbers and a ,b ,c be real numbers not all zero, such that a+b+c=0a n da z_1+b z_2+c z_3=0. Show that z_1, z_2,z_3 are collinear.

If z_1a n dz_2 are two nonzero complex numbers such that = |z_1+z_2|=|z_1|+|z_2|, then a rgz_1-a r g z_2 is equal to -pi b. pi/2 c. 0 d. pi/2 e. pi