******************************* Complex Numbers Problems Part 3 ******************************* 101. Two different non-parallel lines cut the circle :math:`|z|= r` at points :math:`a, b, c, d` respectively. Prove that these two lines meet at point given by :math:`\frac{a^{-1} + b^{-1} + c^{-1} + d^{-1}}{a^{-1}b^{-1}c^{-1}d^{-1}}`. 102. If :math:`z = 2 + t + i\sqrt{3 - t^2},` where :math:`t` is real and :math:`t^2 < 3,` show that the modulus of :math:`(z+1)(z-1)` is independent of :math:`t.` Also, show that the locus of the point :math:`z` for different value of :math:`t` is a circle and find its center and radius. 103. Let :math:`z_1, z_2, z_3` be the three non-zero complex numbers such that :math:`z_2 \ne 1, a = |z_1|, b = |z_2|` and :math:`c = |z_3|.` .. math:: \text{Let } \begin{vmatrix} a & b & c\\ b & c & a\\ c & a & b \end{vmatrix} = 0, then show that :math:`arg\left( \frac{z_3}{z_2}\right) = arg\left( \frac{z_3 - z_1}{z_2 - z_1}\right)^2.` 104. :math:`P` is such a point that on a circle with :math:`OP` as diameter two points :math:`Q` and :math:`R` are taken such that :math:`\angle POQ = \angle QOR = \theta.` If :math:`O` is the origin and :math:`P, Q \text{ and }R` are represented by the complex numbers :math:`z_1, z_2 \text{ and } z_3` respectively, show that :math:`z_2^2 cos2\theta = z_1z_3 cos^2\theta.` 105. Find the equation in complex variables of all circles which are orthogonal to :math:`|z| = 1` and :math:`|z - 1| = 4.` 106. Find the real values real value of the parameter :math:`t` for which there is at least one complex number :math:`z = x + iy` satisfying the condition :math:`|z+3| = t^2 - 2i + 6` and the inequality :math:`z - 3\sqrt{3}i < t^2.` 107. If :math:`a, b, c` and :math:`d` are real and :math:`ad > bc,` show that the imaginary parts of the complex number :math:`z` and :math:`\frac{az + b}{cz + d}` have the same sign. 108. If :math:`z_1 = x_1 + iy_1, z_2 = x_2 + iy_2` and :math:`z_1 = \frac{i(z_2 + 1)}{z_2 - 1}` prove that .. math:: x_1^2 + y_1^2 - x_1 = \frac{x_2^2 + y_2^2 + 2x_2 - 2y_2 +1}{(x_2 - 1)^2 + y_2^2} 109. Simplify the following: .. math:: \frac{(cos3\theta - isin3\theta)^6(sin\theta - icos\theta)^3}{(cos2\theta + isin2\theta)^5} 110. Find all complex numbers such that :math:`z^2 + |z| = 0.` 111. Solve the equation :math:`z^2 + z|z| + |z^2| = 0.` 112. Solve the equation :math:`2z = |z| + 2i` in complex numbers. 113. If :math:`a > 0` and :math:`z|z| + az + 1 = 0`, show that :math:`z` is a negative real number. 114. For every real number :math:`a > 0` find all complex numbers :math:`z` satisfying the equation :math:`z|z| + az + i = 0` 115. For every real number :math:`a > 0` determine the complex numbers :math:`z` which will satisfy the equation :math:`|z|^2 -2iz + 2a(1 + i) = 0` 116. For any two complex numbers :math:`z_1` and :math:`z_2` and any real numbers :math:`a` and :math:`b`, show that :math:`|az_1 -bz_2|^2 + |bz_1 - az_2|^2 = (a^2 + b^2)(|z_1|^2 + |z_2|^2)` 117. If :math:`\alpha` and :math:`\beta` are any two complex numbers, show that :math:`|\alpha + \beta|^2 = |\alpha|^2 + |\beta|^2 + Re(\alpha\overline{\beta}) + Re(\overline{\alpha}\beta)` 118. Find the integral solution of the following equations (i) :math:`(3 + 4i)^x = 5^\frac{x}{2}` (ii) :math:`(1 - i)^x = 2^x` (iii) :math:`(1 - i)^x = (1 + i)^x` 119. Prove that :math:`|1 - \overline{z_1}z_2|^2 - |z_1 - z_2|^2 = (1 - |z_1|^2)(1 - |z_2|^2)` 120. If :math:`a_i, b_i \in R, i = 1, 2, 3, ..., n` show that .. math:: \left(\sum_{n=1}^na_i\right)^2 + \left(\sum_{n=1}^nb_i\right)^2 \le \left(\sum_{n=1}^n\sqrt{a_i^2 + b_i^2}\right)^2 121. Let :math:`\left|\frac{\overline{z_1} - 2\overline{z_2}}{2 - z_1\overline{z_2}}\right| = 1` and :math:`|z_2| \ne 1,` where :math:`z_1` and :math:`z_2` are complex numbers, show that :math:`|z_1| = 2.` 122. If :math:`|z_1| < 1` and :math:`\left|\frac{z_1 - z_2}{1 - \overline{z_1}z_2}\right| < 1,` then show that :math:`|z_2| < 1` 123. If :math:`z_1` and :math:`z_2` are complex numbers and :math:`u = \sqrt{z_1z_2},` prove that .. math:: |z_1| + |z_2| = \left|\frac{z_1 + z_2}{2} + u\right| + \left|\frac{z_1 + z_2}{2} - u\right| 124. If :math:`z_1` and :math:`z_2` are the roots of the equation :math:`\alpha z^2 + 2\beta z + \gamma = 0,` then prove that :math:`|\alpha||(|z_1| + |z_2|) = |\beta + \sqrt{\alpha \gamma}| + |\beta - \sqrt{\alpha \gamma}|` 125. If :math:`a, b, c` are complex numbers such that :math:`a + b + c = 0` and :math:`|a| = |b| = |c| = 1,` find the value of :math:`\frac{1}{a} + \frac{1}{b} + \frac{1}{c}.` 126. If :math:`|z + 4| \le 3,` find the least and greatest value of :math:`|z + 1|.` 127. Show that for any two non-zero complex numbers :math:`z_1, z_2` .. math:: (|z_1| + |z_2|)\left(\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right) \le 2|z_1 + z_2| 128. Show that the necessary and sufficient condition for both the roots of the equation :math:`z^2 + az + b = 0` to be of unit modulus are :math:`|a| \le 2, |b| = 1, argb = 2arga` 129. If :math:`z` is a complex number, show that :math:`|z| \le |R(z)| + |I(z)| \le \sqrt{2}|z|`. 130. If :math:`\left|z - \frac{4}{z}\right| = 2` show that the greatest value of :math:`|z|` is :math:`\sqrt{5} + 1.` 131. If :math:`\alpha, \beta, \gamma, \delta` be the real roots of the equation :math:`ax^4 + bx^3 + cx^2 + dx + e = 0,` show that :math:`a^2(1 + \alpha^2)(1 + \beta)^2(1 + \gamma)^2(1 + \delta)^2 = (a - c + e)^2 + (b - d)^2.` 132. If :math:`a_i \in R, i = 1, 2, ..., n` and :math:`\alpha_1, \alpha_2, ..., \alpha_n` are the roots of the equation .. math:: x^n + a_1x^{n - 1} + a_2x^{n - 2} + ... + a_{n - 1}x + a)n = 0, show that .. math:: \prod_{i = 1}^n(1 + \alpha_i^2) = (1 - a_2 + a_4 - ...)^2 + (a_1 - a_3 + ...)^2 133. If the complex numbers :math:`z_1, z_2, z_2` are the vertices of an equilateral triangle such that :math:`|z_1| = |z_2| = |z_3|,` prove that :math:`z_1 + z_2 + z_3 = 0.` 134. If :math:`z_1 + z_2 + z_3 = 0` and :math:`|z_1| = |z_2| = |z_3| = 1,` then prove that the points :math:`z_1, z_2, z_3` are the vertices of an equilateral triangle inscribed in an unit circle. 135. If :math:`z_1, z_2, z_3` be the vertices of of an equilateral triangle in the Argand plane whose circumcenter is :math:`z_0` then prove that :math:`z_1^2 + z_2^2 + z_3^2 = 3z_0^2.` 136. Prove that the complex numbers :math:`z_1` and :math:`z_2` and the origin form an equilateral triangle if :math:`z_1^2 + z_2^2 - z_1z_2 = 0.` 137. If :math:`z_1` and :math:`z_2` be the roots of the equation :math:`z^2 + az + b = 0,` then prove that the origin, :math:`z_1` and :math:`z_2` form an equilateral triangle if :math:`a^2 = 3b.` 138. Let :math:`z_1, z_2` and :math:`z_3` be the roots of the equation :math:`z^3 + 3\alpha z^2 + 3\beta z + \gamma = 0,` where :math:`\alpha, \beta` and :math:`\gamma` are complex numbers and that these represent the vertices of :math:`A, B` and :math:`C` of a triangle. Find the centroid of :math:`\triangle ABC.` Show that the triangle will be equilateral, if :math:`\alpha^2 = \beta.` 139. If :math:`z_1, z_2` and :math:`z_3` are in A.P., then prove that they are collinear. 140. If :math:`z_1, z_2` and :math:`z_3` are collinear points in Argand plane then show that one of the following holds .. math:: -z_1|z_2 - z_3| + z_2|z_3 - z_1| + z_3|z_1 - z_2| = 0 z_1|z_2 - z_3| - z_2|z_3 - z_1| + z_3|z_1 - z_2| = 0 z_1|z_2 - z_3| + z_2|z_3 - z_1| - z_3|z_1 - z_2| = 0 141. Find the locus of point :math:`z` if :math:`\frac{z - i}{z + i}` is purely imaginary. 142. What region in the Argand plane is represented by the inequality :math:`1 < |z - 3 -4i| < 2.` 143. Find the locus of point :math:`z` if :math:`|z - 1| + |z + 1| \le 4.` 144. If :math:`z = t + 5 + i\sqrt{4 -t^2}` and :math:`t` is real, find the locus of :math:`z.` 145. If :math:`\frac{z^2}{z - 1}` is real show that locus of :math:`z` is a circle with center :math:`(1, 0)` and radius unity and the :math:`x`-axis. 146. If :math:`|z^2 - 1| = |z|^2 + 1,` show that locus of :math:`z` is a straight line. 147. Find the locus of point :math:`z` if :math:`\frac{\pi}{3} \le arg~z \le \frac{3\pi}{2}.` 148. Find the locus of the point :math:`z` if :math:`arg\left(\frac{z - 2}{z + 2}\right) = \frac{\pi}{3}.` 149. Show that the locus of the point :math:`z` satisfying the condition :math:`arg\left(\frac{z - 1}{z + 1}\right) = \frac{\pi}{2}` is the semicircle above :math:`x`-axis whose diameter is the joins of the points :math:`(-1, 0)` and :math:`(1, 0)` excluding those points. 150. Find the locus of the point :math:`z` if :math:`\log_{\sqrt{3}}\frac{|z|^2 - |z| + 1}{2 + |z|} < 2.`