*******************************
Complex Numbers Problems Part 2
*******************************
51. If equation :math:`z^2+\alpha z+\beta = 0` has a real root then prove that
    :math:`(\alpha\overline{\beta}-\beta\overline{\alpha})(\overline{\alpha}-
    \alpha)=(\beta-\overline{\beta})^2`

52. Dividing :math:`f(z)` by :math:`z-i,` we get :math:`i` as remainder and if
    we divide by :math:`z+i,` we get :math:`1+i` as remainder. Find the remainder
    upon division of :math:`f(z)` by :math:`z^2+1`

53. If :math:`iz^3+z^2-z+i=0,` then show that :math:`|z|=1`

54. If :math:`\alpha` and :math:`\beta` are any two complex numbers then show
    that :math:`|\alpha+\sqrt{\alpha^2-\beta^2}| +|\alpha-\sqrt{\alpha^2-\beta^2}|
    = |\alpha+\beta|+|\alpha-\beta|`

55. If :math:`z_1=a+ib` and :math:`z_2=c+id` are complex numbers such that
    :math:`|z_1|=|z_2|=1` and :math:`Re(z_1\overline{z_2})=0` then show that the
    pair of complex numbers :math:`\omega_1=a+ic` and :math:`\omega_2=b+id` satisfy
    (i) :math:`|\omega_1|=|\omega_2|=1` (ii)
    :math:`Re(\omega_1\overline{\omega_2})=0`

56. Prove that :math:`\left|\frac{z_1-z_2}{1-\overline{z_1z_2}}\right|<1` if
    :math:`|z_1|<1, |z_2|<1`

57. Let :math:`z_1=10+6i` and :math:`z_2=4+6i.` If :math:`z` is any complex
    number such that :math:`\frac{z-z_1}{z-z_2}=\frac{\pi}{4},` then prove that
    :math:`|z-7-9i|=3\sqrt{2}.`

58. Find all complex numbers :math:`z` for which
    :math:`arg\left(\frac{3z-6-3i}{2z-8-6i}\right)=\frac{\pi}{4}` and :math:`|z -
    3+i=3|`

59. If :math:`z\le 1, |w|\le 1,` show that :math:`|z - w|^2\le (|z| - |w|)^2 +
    (Arg(z) - Arg(w))^2`

60. If :math:`z` is any non-zero complex number, show that

    (i) :math:`\left|\frac{z}{|z|} - 1\right| \le |arg z|` (ii) :math:`|z-1| \le
    ||z| - 1|+ |z| |arg z|`

61. If :math:`\left|z+\frac{1}{z}\right|=a,` where :math:`z` is a complex
    number and :math:`a>0,` find the greatest and least value of :math:`|z|.`

62. If :math:`z_1, z_2` are complex numbers and :math:`c` is a positive number
    prove that :math:`|z_1+z_2|^2 < (1+c)|z_1|^2 + \left(1+\frac{1}{c}\right)
    |z_2|^2`

63. Let :math:`z_1, z_2` be any two complex numbers and :math:`a,b` be two real
    numbers such that :math:`a^2+b^2 \ne 0.` Prove that

    .. math::
       |z_1|2 + |z_2|^2 - |z_1^2 + z_2^2| \le 2\frac{|az_1+bz_2|^2}{a^2+b^2} \le
       |z_1|^2 + |z_2|^2 + |z_1^2 + z_2^2|

64. If :math:`b+ic=(1+a)z` and :math:`a^2+b^2+c^2=1,` prove that
    :math:`\frac{a+ib}{1+c}=\frac{1+iz}{1-iz},` where :math:`a,b,c` are real
    numbers and :math:`z` is a real number.

65. If :math:`a,b,c, ..., k` are all :math:`n` real roots of the equation
    :math:`x^n + p_1x^{n-1}+p_2x^{n-2} + ... + p_{n-1}x + p_n=0,`
    where :math:`p_1,p_2, ..., p_n` are real, show that
    :math:`(1+a^2)(1+b^2) ... (1+k^2) = (1-p_2+p_4)^2 + (p_1-p_3+ ...)^2`

66. If :math:`f(x) = x^4-8x^3+4x^2+4x+39` and :math:`f(3+2i) = a+ib,` find
    :math:`a:b`

67. If :math:`z_1, z_2, z_3` be the vertices of an equilateral triangle, show
    that

    .. math::
       \frac{1}{z_1-z_2} + \frac{1}{z_2-z_3} + \frac{1}{z_3-z_1}=0 \text{ or }
       z_1^2+z_2^2+z_3^2 = z_1z_2+z_2z_3+z_3z_1

68. If :math:`z_1^2+z_2^2-2z_1z_2cos\theta,` show that the points
    :math:`z_1,z_2` and the origin are the vertices of an isosceles triangle.

69. Let :math:`A` and :math:`B` be two complex numbers such that
    :math:`\frac{A}{B} + \frac{B}{A}=1,` prove that the triangle formed by origin
    and these two points is equilateral.

70. Prove that the area of triangle formed by three complex numbers :math:`z_1,
    z_2, z_3` is

    .. math::
       \left|\sum\frac{(z_2-z_3)|z_1|^2}{4iz_1}\right|

71. If :math:`n>1` then show that the roots of the equation :math:`z^n=(z+1)^n`
    are collinear.

72. If :math:`A, B, C, \text{ and } D` are four complex number then
    show that :math:`AD.BC\le BD.CA + CD.AB`

73. If :math:`a,b\in R \text{ and } a,b\ne 0` then show that the equation of
    line joining these :math:`a` and :math:`ib` is

    .. math::
       \left(\frac{1}{2a}-\frac{i}{2b}\right)z+
       \left(\frac{1}{2a}+\frac{i}{2b}\right)\overline{z} = 1.
    
74. If :math:`z_1` and :math:`z_2` are two complex number such that
    :math:`|z_1| = |z_2| + |z_1-z_2|` then show that :math:`arg~z_1 - arg~z_2 = 2n\pi`
    where :math:`n\in I`

75. If :math:`z=z_1.z_2. ... .z_n,` prove that :math:`arg~z - (arg~z_1+arg~z_2+
    ... + arg~z_n)=2n\pi` where :math:`n\in I`

76. Let :math:`A, B, C, D, E` be points in the complex plane representing the
    complex numbers :math:`z_1, z_2 ,z_3 ,z_4, z_5` respectively. If :math:`(z_3 -
    z_2)z_4 = (z_1 - z_2)z_5,` prove that :math:`\triangle ABC` and :math:`\triangle
    DOE` are similar.

77. Let :math:`z` and :math:`z_0` are two complex numbers and the numbers
    :math:`z, z_0, z\overline{z_0}, 1` are represented by points :math:`P, P_0, Q,
    A` respectively. If :math:`|z|=1,` show that the triangle :math:`POP_0` and :math:`AOQ`
    are congruent or :math:`|z-z_0|=|z\overline{z_0}-1|` where :math:`O` represents
    origin.

78. If the line segment joining :math:`z_1` and :math:`z_2` is divided by
    :math:`P` and :math:`Q` in the ratio of :math:`a:b` internally and
    externally then find :math:`OP^2 + OQ^2`
    where :math:`O` is origin.

79. Let :math:`z_1, z_2, z_3` be three complex numbers and :math:`a, b
    ,c` be real numbers not all zero such that :math:`a + b + c=0` and
    :math:`az_1 + bz_2 + cz_3 = 0,` then show that :math:`z_1, z_2,
    z_3` are collinear. 

80. If :math:`z_1 + z_2 + ... +z_n = 0,` prove that if a line passes through
    origin then all these do not lie on the same side of line provided the points
    do not lie on the line.

81. Suppose the points :math:`z_1, z_2, ..., z_2 (z_i \ne 0)` all lie on one
    side of a line drawn through origin of the complex plane. Prove that the same
    if true of the points :math:`\frac{1}{z_1}, \frac{1}{z_2}, ..., \frac{1}{z_n}.`
    Moreover, show that

    .. math::
       z_1 + z_2 + ... + z_n \ne 0 \text{ and } \frac{1}{z_1} + \frac{1}{z_2} +
       ... + \frac{1}{z_2} \ne 0

82. The points :math:`z_1 = 9+ 12i` and :math:`z_2 = 6 - 8i` are given on a
    complex plane. Find the equation of the bisector of the angle formed by the
    vector representing :math:`z_1` and :math:`z_2.`

83. If the vertices of a triangle :math:`ABC` are :math:`z_1, z_2, z_3,` then
    show that the orthocenter of the :math:`\triangle ABC` is

    .. math::
       \frac{(a secA)z_1 + (b secB)z_2 + (c secC)z_3}{a secA + b secB + c secC}
       \text{ or } \frac{z_1 tanA + z_2 tanB + z_3 tanC}{tanA + tanB + tanC} \text{
       or } z = \frac{\sum z_1^2(\overline{z_2} - \overline{z_3}) + \sum |z_1|^2(z2
       - z_3)}{\sum (z_1\overline{z_2} - z_2\overline{z_1})}

    where :math:`z` is orthocenter and :math:`a, b, c` are sides.

84. If the vertices of a triangle :math:`ABC` are :math:`z_1, z_2, z_3,` then
    show that the circumcenter of the :math:`\triangle ABC` is

    .. math::
       \frac{z_1 sin2A + z_2 sin2B + z_2 sin2C}{sin2A + sin2B + sin2C} \text{ or }
       z = \frac{\sum z_1\overline{z_1}z_2 - z_3}{\sum \overline{z_1}(z_2 - z_3)}

    where :math:`z` is circumcenter.

85. :math:`ABCD` is a rhombus described in clockwise direction, vertices are
    given by :math:`z_1, z_2, z_3, z_4` respectively and :math:`\angle CBA = 2\pi /
    3.` Show that :math:`2\sqrt{3}z_2 = (\sqrt{3} - i)z_1 + (\sqrt{3} + i)z_3` and 
    :math:`2\sqrt{3}z_4 = (\sqrt{3} - i)z_3 + (\sqrt{3} + i)z_1`

86. The point :math:`A, B, C` represent the complex numbers :math:`z_1, z_2,
    z_3` respectively and the angles of the triangle :math:`ABC` at :math:`B` and
    :math:`C` are both :math:`\frac{1}{2}(\pi - \alpha)` then prove that
    :math:`(z_3 - z_2)^2 = 4` and :math:`(z_3 - z_1)(z_1 - z_2) sin^2\left(
    \frac{\alpha}{2}\right)`

87. Points :math:`z_1` and :math:`z_2` are adjacent points of a regular polygon
    with :math:`n` sides. If :math:`z_3` is adjacent vertex to :math:`z_2`
    where :math:`z_2 \ne z_1` then find :math:`z_3.`
    
88. If :math:`z_1, z_2` and :math:`z_3` are sides of an equilateral triangle and
    :math:`z_0` is the centroid then prove that :math:`z_1^2 + z_2^2 + z_3^2 =
    3z_0^2.` 

89. Let :math:`A_1, A_2, ..., A_n` are vertices of an :math:`n` sided polygon
    such that :math:`\frac{1}{A_1A_2} = \frac{1}{A_1A_3} + \frac{1}{A_2A_4},`
    find the value of :math:`n.`

90. If :math:`A_1, A_2, ..., A_n` be the vertices of a regular polygon of
    :math:`n` sides in a circle of radius unity. Find the values of
    :math:`|A_1A_2|^2 + |A_1A_3|^2 + ... + |A_1A_n|^2` and
    :math:`|A_1A_2||A_1A_3| ... |A_1A_n|`

91. If :math:`|z|=2` the show that points representing the complex numbers
    :math:`-1 + 5z` lie on a circle.

91. If :math:`z-6-8i \le 4` then find the greatest and least value of :math:`z.`

92. If :math:`z-25i \le 15` then find the least positive value of :math:`argz.`

93. Show that the equation :math:`|z - z_1|^2 + |z-z_2|^2 = k` where :math:`k
    \in R` will represent a circle if :math:`k \ge \frac{1}{2}|z_1 - z_2|^2.`
94. If :math:`|z-1| = 1,` prove that :math:`\frac{z-2}{z} = i tan(rag z).`

95. Find the locus of :math:`z` if :math:`arg\left( \frac{z - 1}{z + 1}\right)
    = \frac{\pi}{4}.`

96. If :math:`\alpha` is real and :math:`z` is a complex number and :math:`u`
    and :math:`v` be the real and imaginary parts of :math:`(z-1)(cos\alpha - i
    sin\alpha) + (z - 1)^{-1}(cos\alpha + i sin \alpha).` Prove that the locus
    of the points representing the complex numbers such that :math:`v=0` is a
    circle of unit radius with center at point :math:`(1, 0)` and a straight
    line through the center of the circle.

97. If :math:`|a_n| < 2` for :math:`n = 1, 2, 3, ...` and :math:`1 + a_1z +
    a_2z^2 + ... + a_nz^n = 0` show that :math:`z` does not lies in the
    interior of the circle :math:`|z| = \frac{1}{3}.`

98. Show that the roots of the equation :math:`z^n cos\theta_0 + z^{n-1}
    cos\theta_1 + ... + cos\theta_n = 2` where :math:`\theta_0 + \theta_1 +
    ... + \theta_n \in R` lies outside the circle :math:`|z| = \frac{1}{2}.`

99. :math:`z_1, z_2, z_3` are non-zero, non-collinear complex numbers such that
    :math:`\frac{2}{z_1} = \frac{1}{z_2} + \frac{1}{z_3}` show that :math:`z_1,
    z_2, z_3` lie on a circle passing through the origin.

100. :math:`A, B, C` are the points representing the complex numbers
     :math:`z_1, z_2, z_3` respectively on the complex plane and the
     circumcenter of the triangle :math:`ABC` lies on the origin. If the
     altitude of the triangle through the vertex :math:`A` meets the
     circle again at :math:`P,` prove that :math:`P` represents the complex
     number :math:`\frac{z_2z_3}{z_1}.`