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.`

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)

Find the complex number z satisfying R e(z^2) =0,|z|=sqrt(3.)

If z_1a n dz_2 are two complex numbers and c >0 , then prove that |z_1+z_2|^2lt=(1+c)|z_1|^2+(1+c^(-1))|z_2|^2dot

Find the Area bounded by complex numbers arg|z|lepi/4 and |z-1|lt|z-3|

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

If z_1a n dz_2 are complex numbers and u=sqrt(z_1z_2) , then prove that |z_1|+|z_2|=|(z_1+z_2)/2+u|+|(z_1+z_2)/2-u|

If z and w are two complex numbers simultaneously satisfying te equations, z^3+w^5=0 and z^2 +overlinew^4 = 1, then

If z_1, z_2, z_3 are three nonzero complex numbers such that z_3=(1-lambda)z_1+lambdaz_2w h e r elambda in R-{0}, then prove that points corresponding to z_1, z_2a n dz_3 are collinear .

If |z-1|lt=2a n d|omegaz-1-omega^2|=a where omega is cube root of unity , then complete set of values of a is a. 0lt=alt=2 b. 1/2lt=alt=(sqrt(3))/2 c. (sqrt(3))/2-1/2lt=alt=1/2+(sqrt(3))/2 d. 0lt=alt=4

If z_1 and z_2 are two complex number such that |z_1|<1<|z_2| then prove that |(1-z_1 bar z_2)/(z_1-z_2)|<1