******************************* Complex Numbers Problems Part 4 ******************************* 151. If :math:`O` be the center of the circle circumscribing the equilateral :math:`\triangle ABC` and its radius be unity and :math:`A` lies on the :math:`x`-axis. Find the complex numbers represented by :math:`B` and :math:`C.` 152. :math:`ABCD` is a rhombus. Its diagonals :math:`AC` and :math:`BD` intersect at a point :math:`M` and satisfy :math:`BD = 2AC.` If the points :math:`D` and :math:`M` represent the complex numbers :math:`1 + i` and :math:`2 - i` respectively, then find the complex number represented by :math:`A`. 153. If :math:`z_1, z_2, z_3` and :math:`z_4` are the vertices of a square taken in anticlockwise order, prove that :math:`z_3 = -iz_1 + (1 + i)z_2` and :math:`z_4 = (1 - i)z_1 + iz_2.` 154. Let :math:`z_1, z_2` and :math:`z_3` are the vertices of an equilateral triangle in the circle :math:`|z| = 2.` If :math:`z_1 = 1 + i\sqrt{3},` then find :math:`z_2` and :math:`z_3.` 155. If :math:`a` and :math:`b` are real numbers between 0 and 1 such that the points :math:`z_1 = a + i, z_2 = 1 + bi,` and :math:`z_3 = 0` form an equilateral triangle then find :math:`a` and :math:`b.` 156. Let :math:`ABCD` be a square described in the anticlockwise sense in the Argand plane. If :math:`A` represents :math:`(3 + 5i)` and the center of the square represents :math:`\left(\frac{7}{2} + \frac{5}{2}i\right).` Find the numbers represented by :math:`B, C` and :math:`D`. 157. Find the vertices of a regular polygon of :math:`n` sides, if its center is located at origin and one of its vertices is :math:`z_1.` 158. Prove that the points :math:`a(cos\alpha + i sin\alpha), b(cos\beta + i sin\beta)` and :math:`c(cos\gamma + i sin\gamma)` in the Argand plane are collinear if :math:`bcsin(\beta - \gamma) + casin(\gamma - \alpha) + absin(\alpha -\beta) = 0.` 159. :math:`A` represents the number :math:`6i, B` the number :math:`3` and :math:`P` the complex number :math:`z`. If :math:`P` moves such that :math:`PA : PB = 2 : 1,` show that :math:`z\overline{z} = (4 + 2i)z + (4 - 2i)\overline{z}.` Also, show that the locus of P is a circle, find its radius and center. 160. Show that if the points :math:`z_1, z_2, z_3` and :math:`z_4` taken in order are concyclic then the expression :math:`\frac{(z_3 - z_1)(z_4 - z_2)}{(z_3 - z_2)(z_4 -z_1)}` is purely real. 161. Let :math:`z_1, z_2, z_3` and :math:`z_4` be the vertices of a quadrilateral. Prove that quadrilateral is cyclic if :math:`z_1z_2 + z_3z_4 = 0` and :math:`z_1 + z_2 = 0.` 162. Show that the triangles whose vertices are :math:`z_1, z_2, z_3` and :math:`z_1', z_2', z_3'` are similar if .. math:: \begin{vmatrix}z_1 & z_1' & 1\\z_2 & z_2' & 1\\z_3 & z_3' & 1\end{vmatrix} = 0 163. If :math:`a, b, c` and :math:`u, v, \omega` are the complex numbers representing two triangles such that :math:`c = (1 - r)a + rb` and :math:`\omega = (i - r)u + rv,` where :math:`r` is a complex number, prove that the two triangles are similar. 164. Find the equation of perpendicular bisector of the line segment joining points :math:`z_1` and :math:`z_2.` 165. Find the equation of the circle having the line segment joining :math:`z_1` and :math:`z_2` as diameter. 166. If :math:`\left|\frac{z - z_1}{z - z_2}\right| = c, c \ne~0,` then show that the locus of :math:`z` is a circle. 167. If :math:`|z| = 1,` find the locus of the point :math:`\frac{2}{z}.` 168. If for any two complex numbers :math:`z_1` and :math:`z_2, |z_1 + z_2| = |z_1| + |z_2|` prove that :math:`arg(z_1) - arg(z_2) = 2n\pi.` 169. Find the complex number :math:`z` the least in absolute value which satisfies the condition :math:`|z - 2 + 2i| = 1.` 170. Find the point in the first quadrant, on the curve :math:`|z - 5i| = 3` whose argument is minimum. 171. Find the set of points of the coordinate plane which satisfy the inequality .. math:: log_\frac{1}{2}\left(\frac{|z - 1| + 4}{3|z - 1| - 2}\right) > 1 172. Find the set of all points on the :math:`xy`-plane whose coordinates satisfy the following conditions: the number :math:`z^2 + z + 1` is real and positive. 173. Find the real values of the parameter :math:`a` for which at least one complex number :math:`z` satisfies the equality :math:`|z - az| = a + 4` and the inequality :math:`|z - 2| < 1.` 174. Find the real values of parameter :math:`t` for which at least one complex number :math:`z` satisfies the equality :math:`|z + \sqrt{2}| = t^2 - 3t +2` and the inequality :math:`|z + t\sqrt{2}| < t^2.` 175. Among the complex numbers which satisfy the condition :math:`|z - 15i| \le 15,` find the number having least positive argument. 176. Find :math:`z` for which :math:`|z - 5i| \le 1` and :math:`arg(z)` is minimum. 177. Find the real value of :math:`a` for which there is at least one complex number satisfying :math:`|z + 4i| = \sqrt{a^2 - 12a + 28}` and :math:`|z - 4\sqrt{3}| < a.` 178. Find the set of points belonging to the coordinate plane :math:`xy,` for which the real part of complex number :math:`(1 + i)z^2` is positive. 179. Solve the equation :math:`2z = |z| + 2i` in complex numbers. 180. Show that all the roots of the equation :math:`a_1z^3 + a_2z^2 + a_3z + a_4 = 3,` where :math:`|a_i| \le 1, i= 1, 2, 3, 4` lie outside the circle with center as origin and radius :math:`\frac{2}{3}.` 181. Given that :math:`\sum_{i = 1}^nb_i = 0` and :math:`\sum_{i = 1}^nb_iz_i = 0` where :math:`b_i`'s are non-zero real numbers, no three of :math:`z_i`'s form a straight line. Prove that :math:`z_i`'s are concyclic if :math:`b_1b_2|z_2 - z_2|^2 = b_3b_4|z_3 - z_4|^2.` 182. A cubic equation :math:`f(x) = 0` has one real root :math:`\alpha` and two complex roots :math:`\beta + i\gamma.` Points :math:`A, B` and :math:`C` represent these roots respectively. Show that the roots of the derived equation :math:`f'(x) = 0` are complex if :math:`A` falls inside one of the two equilateral triangles described on base :math:`BC`. 183. Prove that reflection of :math:`\overline{a}z + a\overline{z} = 0` in the real axis is :math:`\overline{a}\overline{z} + az = 0.` 184. If :math:`\alpha, \beta, \gamma, \delta` are four complex numbers such that :math:`\frac{\gamma}{\delta}` is real and :math:`\alpha\delta - \beta\gamma \ne 0` then prove that :math:`z = \frac{\alpha + \beta t}{\gamma + \delta t}, t \in \mathbf{R}` represents a straight line. 185. If :math:`\omega, \omega^2` are cube roots of unity, then prove that (i) :math:`(3 + 3\omega + 5\omega^2)^2 - (2 + 6\omega + 2\omega^2) = 0` (ii) :math:`(2 - \omega)(2 - \omega^2)(2 -\omega^{10})(2 - \omega^{11}) = 49` (iii) :math:`(1 - \omega)(1 - \omega^2)(1 - \omega^4)(1 - \omega^5) = 9` (iv) :math:`(1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 = 32` (v) :math:`1 + \omega^n + \omega^{2n} = 3` where :math:`n > 0, n \in \mathbf{I}` and is a multiple of 3. (vi) :math:`1 + \omega^n + \omega^{2n} = 0` where :math:`n > 0, n \in \mathbf{I}` and is not a multiple of 3. 186. Resolve into linear factors :math:`a^2 + b^2 + c^2 -ab -bc -ca` 187. Prove that .. math:: \sqrt{-1 - \sqrt{-1 - \sqrt{-1 - ... to \infty}}} = \omega \text{ or } \omega^2 188. If :math:`x = a + b, y = a\omega + b\omega^2, z = a\omega^2 + b\omega,` prove that :math:`x^3 + y^3 + z^3 = 3(a^3 + b^3)` and :math:`xyz = a^3 + b^3.` 189. Resolve into linear factors: (i) :math:`a^2 -ab + b^2` (ii) :math:`a^2 + ab + b^2` (iii) :math:`a^3 + b^3` (iv) :math:`a^3 - b^3` (v) :math:`a^3 + b^3 +c^3 -3abc` 190. Show that :math:`x^{3p} + x^{3q+1} + x^{3r+2},` where :math:`p, q, r` are positive integers is divisible by :math:`x^2 + x + 1.` 191. Show that :math:`x^{3p} + x^{3q+1} + x^{3r+2} + x^{4s+3},` where :math:`p, q, r, s` are positive integers is divisible by :math:`x^3 + x^2 + x + 1.` 192. If :math:`p = a + b + c, q = a + b\omega + c\omega^2, r = a + b\omega^2 + c\omega,` where :math:`\omega` is a cube root of unity, prove that :math:`p^3 + q^3 + r^3 - 3pqr = 27abc.` 193. If :math:`\omega` be an imaginary root of unity, prove that :math:`(a + b\omega + c\omega^2)^3 + (a + b\omega^2 + c\omega)^3 = (2a - b -c)(2b -a -c)(2c -a -b).` 194. If :math:`ax + cy + bz = X, cx + by + az = Y, bx + ay + cz = Z` show that (i) :math:`(a^2 + b^2 + c^2 - ab - bc - ca)(x^2 + y^2 + z^2 - xy - yz -zx) = X^2 + Y^2 + Z^2 - XY -YZ -ZY` (ii) :math:`(a^3 + b^3 - c^3 - 3abc)(x^3 + y^3 + z^3 -3xyz) = X^3 + Y^3 + Z^3 - 3XYZ` 195. Prove that .. math:: \left(\frac{cos\theta + isin\theta}{sin\theta + icos\theta}\right)^4 = cos8\theta + isin8\theta 196. If :math:`z^2 -2zcos\theta +1 =0,` show that :math:`z^2 + z^{-2} = 2cos2\theta` 197. If :math:`x_r = cos\frac{\pi}{2^r} + isin\frac{\pi}{2^r},` prove that :math:`x_1x_2x_3 ... to \infty = -1.` 198. Prove that :math:`(1 + i)^n + (1 - i)^n = 2^{\frac{n}{2} + 1}cos\frac{n\pi}{4}.` 199. Show that the value of .. math:: \sum_{k = 1}^6\left(sin\frac{2\pi k}{7} -icos\frac{2\pi k}{7}\right) is :math:`i.` 200. Show that .. math:: e^{2mi cot^{-1}p}\left(\frac{pi + 1}{pi - 1}\right)^m = 1