Let `z=9+b i ,w h e r eb` is nonzero real and `i^2=-1.` If the imaginary part of `z^2a n dz^3` are equal, then `b/3` is ______.

Let z be a complex number such that the imaginary part of z is nonzero and a = z^2 + z + 1 is real. Then a cannot take the value

Prove that the locus of midpoint of line segment intercepted between real and imaginary axes by the line bara z + a bar z+b=0,w h e r eb is a real parameterand a is a fixed complex number with nondzero real and imaginary parts, is a z+ bara barz =0.

The locus of point z satisfying R e(1/z)=k ,w h e r ek is a nonzero real number, is a. a straight line b. a circle c. an ellipse d. a hyperbola

Consider the equation 10 z^2-3i z-k=0,w h e r ez is a following complex variable and i^2=-1. Which of the following statements ils true? (a) For real complex numbers k , both roots are purely imaginary. (b) For all complex numbers k , neither both roots is real. (c) For all purely imaginary numbers k , both roots are real and irrational. (d) For real negative numbers k , both roots are purely imaginary.

Let z_1a n dz_2 be complex numbers such that z_1!=z_2 and |z_1|=|z_2|dot If z_1 has positive real part and z_2 has negative imaginary part, then (z_1+z_2)/(z_1-z_2) may be (a)zero (b) real and positive (c) real and negative (d) purely imaginary

Let |z|=2a n dw-(z+1)/(z-1),w h e r ez ,w , in C (where C is the set of complex numbers). Then product of least and greatest value of modulus of w is__________.

Show that the equation of a circle passing through the origin and having intercepts aa n db on real and imaginary axes, respectively, on the argand plane is given by z bar z =a a(R e z)+b(I mz)dot

Given z is a complex number with modulus 1. Then the equation [(1+i a)/(1-i a)]^4=z has (a) all roots real and distinct (b)two real and two imaginary (c) three roots two imaginary (d)one root real and three imaginary

Let a ,b ,c be real. If a x^2+b x+c=0 has two real roots alphaa n dbeta,w h e r ealpha >1 , then show that 1+c/a+|b/a|<0