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

Find the non-zero complex number z satisfying z =i z^2dot

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

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

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

Find a complex number z satisfying the equation z+sqrt(2)|z+1|+i=0.

Find a complex number z satisfying the equation z+sqrt(2)|z+1|+i=0.

Find a complex number z satisfying the equation z+sqrt(2)|z+1|+i=0.

Statement-1 : IF |z+1/z| =a , where z is a complex number and a is a real number, the least and greatest values of |z| are (sqrt(a^(2)+4-a))/2 and (sqrt(a^(2)+ 4)+a)/2 and Statement -2 : For a equal ot zero the greatest and the least values of |z| are equal .

If z_1 and z_2 are two complex numbers such that |z_1|lt1lt|z_2| then prove that |(1-z_1barz_2)/(z_1-z_2)|lt1

If z_(1) and z_(2) are two complex numbers such that |z_(1)| lt 1 lt |z_(2)| , then prove that |(1- z_(1)barz_(2))//(z_(1)-z_(2))| lt 1