21. Complex Numbers Problems Part 3

101. Two different non-parallel lines cut the circle \(|z|= r\) at points \(a, b, c, d\) respectively. Prove that these two lines meet at point given by \(\frac{a^{-1} + b^{-1} + c^{-1} + d^{-1}}{a^{-1}b^{-1}c^{-1}d^{-1}}\).

102. If \(z = 2 + t + i\sqrt{3 - t^2},\) where \(t\) is real and \(t^2 < 3,\) show that the modulus of \((z+1)(z-1)\) is independent of \(t.\) Also, show that the locus of the point \(z\) for different value of \(t\) is a circle and find its center and radius.

103. Let \(z_1, z_2, z_3\) be the three non-zero complex numbers such that \(z_2 \ne 1, a = |z_1|, b = |z_2|\) and \(c = |z_3|.\)

     \[\text{Let } \begin{vmatrix} a & b & c\\ b & c & a\\ c & a & b \end{vmatrix} = 0, \]

     then show that \(arg\left( \frac{z_3}{z_2}\right) = arg\left( \frac{z_3 - z_1}{z_2 - z_1}\right)^2.\)

104. \(P\) is such a point that on a circle with \(OP\) as diameter two points \(Q\) and \(R\) are taken such that \(\angle POQ = \angle QOR = \theta.\) If \(O\) is the origin and \(P, Q \text{ and }R\) are represented by the complex numbers \(z_1, z_2 \text{ and } z_3\) respectively, show that \(z_2^2 cos2\theta = z_1z_3 cos^2\theta.\)

105. Find the equation in complex variables of all circles which are orthogonal to \(|z| = 1\) and \(|z - 1| = 4.\)

106. Find the real values real value of the parameter \(t\) for which there is at least one complex number \(z = x + iy\) satisfying the condition \(|z+3| = t^2 - 2i + 6\) and the inequality \(z - 3\sqrt{3}i < t^2.\)

107. If \(a, b, c\) and \(d\) are real and \(ad > bc,\) show that the imaginary parts of the complex number \(z\) and \(\frac{az + b}{cz + d}\) have the same sign.

108. If \(z_1 = x_1 + iy_1, z_2 = x_2 + iy_2\) and \(z_1 = \frac{i(z_2 + 1)}{z_2 - 1}\) prove that

     \[x_1^2 + y_1^2 - x_1 = \frac{x_2^2 + y_2^2 + 2x_2 - 2y_2 +1}{(x_2 - 1)^2 + y_2^2} \]

109. Simplify the following:

     \[\frac{(cos3\theta - isin3\theta)^6(sin\theta - icos\theta)^3}{(cos2\theta + isin2\theta)^5} \]

110. Find all complex numbers such that \(z^2 + |z| = 0.\)

111. Solve the equation \(z^2 + z|z| + |z^2| = 0.\)

112. Solve the equation \(2z = |z| + 2i\) in complex numbers.

113. If \(a > 0\) and \(z|z| + az + 1 = 0\), show that \(z\) is a negative real number.

114. For every real number \(a > 0\) find all complex numbers \(z\) satisfying the equation \(z|z| + az + i = 0\)

115. For every real number \(a > 0\) determine the complex numbers \(z\) which will satisfy the equation \(|z|^2 -2iz + 2a(1 + i) = 0\)

116. For any two complex numbers \(z_1\) and \(z_2\) and any real numbers \(a\) and \(b\), show that \(|az_1 -bz_2|^2 + |bz_1 - az_2|^2 = (a^2 + b^2)(|z_1|^2 + |z_2|^2)\)

117. If \(\alpha\) and \(\beta\) are any two complex numbers, show that \(|\alpha + \beta|^2 = |\alpha|^2 + |\beta|^2 + Re(\alpha\overline{\beta}) + Re(\overline{\alpha}\beta)\)

118. Find the integral solution of the following equations (i) \((3 + 4i)^x = 5^\frac{x}{2}\) (ii) \((1 - i)^x = 2^x\) (iii) \((1 - i)^x = (1 + i)^x\)

119. Prove that \(|1 - \overline{z_1}z_2|^2 - |z_1 - z_2|^2 = (1 - |z_1|^2)(1 - |z_2|^2)\)

120. If \(a_i, b_i \in R, i = 1, 2, 3, ..., n\) show that

     \[\left(\sum_{n=1}^na_i\right)^2 + \left(\sum_{n=1}^nb_i\right)^2 \le \left(\sum_{n=1}^n\sqrt{a_i^2 + b_i^2}\right)^2 \]

121. Let \(\left|\frac{\overline{z_1} - 2\overline{z_2}}{2 - z_1\overline{z_2}}\right| = 1\) and \(|z_2| \ne 1,\) where \(z_1\) and \(z_2\) are complex numbers, show that \(|z_1| = 2.\)

122. If \(|z_1| < 1\) and \(\left|\frac{z_1 - z_2}{1 - \overline{z_1}z_2}\right| < 1,\) then show that \(|z_2| < 1\)

123. If \(z_1\) and \(z_2\) are complex numbers and \(u = \sqrt{z_1z_2},\) prove that

     \[|z_1| + |z_2| = \left|\frac{z_1 + z_2}{2} + u\right| + \left|\frac{z_1 + z_2}{2} - u\right| \]

124. If \(z_1\) and \(z_2\) are the roots of the equation \(\alpha z^2 + 2\beta z + \gamma = 0,\) then prove that \(|\alpha||(|z_1| + |z_2|) = |\beta + \sqrt{\alpha \gamma}| + |\beta - \sqrt{\alpha \gamma}|\)

125. If \(a, b, c\) are complex numbers such that \(a + b + c = 0\) and \(|a| = |b| = |c| = 1,\) find the value of \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}.\)

126. If \(|z + 4| \le 3,\) find the least and greatest value of \(|z + 1|.\)

127. Show that for any two non-zero complex numbers \(z_1, z_2\)

     \[(|z_1| + |z_2|)\left(\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right) \le 2|z_1 + z_2| \]

128. Show that the necessary and sufficient condition for both the roots of the equation \(z^2 + az + b = 0\) to be of unit modulus are \(|a| \le 2, |b| = 1, argb = 2arga\)

129. If \(z\) is a complex number, show that \(|z| \le |R(z)| + |I(z)| \le \sqrt{2}|z|\).

130. If \(\left|z - \frac{4}{z}\right| = 2\) show that the greatest value of \(|z|\) is \(\sqrt{5} + 1.\)

131. If \(\alpha, \beta, \gamma, \delta\) be the real roots of the equation \(ax^4 + bx^3 + cx^2 + dx + e = 0,\) show that \(a^2(1 + \alpha^2)(1 + \beta)^2(1 + \gamma)^2(1 + \delta)^2 = (a - c + e)^2 + (b - d)^2.\)

132. If \(a_i \in R, i = 1, 2, ..., n\) and \(\alpha_1, \alpha_2, ..., \alpha_n\) are the roots of the equation

     \[x^n + a_1x^{n - 1} + a_2x^{n - 2} + ... + a_{n - 1}x + a)n = 0, \]

     show that

     \[\prod_{i = 1}^n(1 + \alpha_i^2) = (1 - a_2 + a_4 - ...)^2 + (a_1 - a_3 + ...)^2 \]

133. If the complex numbers \(z_1, z_2, z_2\) are the vertices of an equilateral triangle such that \(|z_1| = |z_2| = |z_3|,\) prove that \(z_1 + z_2 + z_3 = 0.\)

134. If \(z_1 + z_2 + z_3 = 0\) and \(|z_1| = |z_2| = |z_3| = 1,\) then prove that the points \(z_1, z_2, z_3\) are the vertices of an equilateral triangle inscribed in an unit circle.

135. If \(z_1, z_2, z_3\) be the vertices of of an equilateral triangle in the Argand plane whose circumcenter is \(z_0\) then prove that \(z_1^2 + z_2^2 + z_3^2 = 3z_0^2.\)

136. Prove that the complex numbers \(z_1\) and \(z_2\) and the origin form an equilateral triangle if \(z_1^2 + z_2^2 - z_1z_2 = 0.\)

137. If \(z_1\) and \(z_2\) be the roots of the equation \(z^2 + az + b = 0,\) then prove that the origin, \(z_1\) and \(z_2\) form an equilateral triangle if \(a^2 = 3b.\)

138. Let \(z_1, z_2\) and \(z_3\) be the roots of the equation \(z^3 + 3\alpha z^2 + 3\beta z + \gamma = 0,\) where \(\alpha, \beta\) and \(\gamma\) are complex numbers and that these represent the vertices of \(A, B\) and \(C\) of a triangle. Find the centroid of \(\triangle ABC.\) Show that the triangle will be equilateral, if \(\alpha^2 = \beta.\)

139. If \(z_1, z_2\) and \(z_3\) are in A.P., then prove that they are collinear.

140. If \(z_1, z_2\) and \(z_3\) are collinear points in Argand plane then show that one of the following holds

     \[-z_1|z_2 - z_3| + z_2|z_3 - z_1| + z_3|z_1 - z_2| = 0 z_1|z_2 - z_3| - z_2|z_3 - z_1| + z_3|z_1 - z_2| = 0 z_1|z_2 - z_3| + z_2|z_3 - z_1| - z_3|z_1 - z_2| = 0\]

141. Find the locus of point \(z\) if \(\frac{z - i}{z + i}\) is purely imaginary.

142. What region in the Argand plane is represented by the inequality \(1 < |z - 3 -4i| < 2.\)

143. Find the locus of point \(z\) if \(|z - 1| + |z + 1| \le 4.\)

144. If \(z = t + 5 + i\sqrt{4 -t^2}\) and \(t\) is real, find the locus of \(z.\)

145. If \(\frac{z^2}{z - 1}\) is real show that locus of \(z\) is a circle with center \((1, 0)\) and radius unity and the \(x\)-axis.

146. If \(|z^2 - 1| = |z|^2 + 1,\) show that locus of \(z\) is a straight line.

147. Find the locus of point \(z\) if \(\frac{\pi}{3} \le arg~z \le \frac{3\pi}{2}.\)

148. Find the locus of the point \(z\) if \(arg\left(\frac{z - 2}{z + 2}\right) = \frac{\pi}{3}.\)

149. Show that the locus of the point \(z\) satisfying the condition \(arg\left(\frac{z - 1}{z + 1}\right) = \frac{\pi}{2}\) is the semicircle above \(x\)-axis whose diameter is the joins of the points \((-1, 0)\) and \((1, 0)\) excluding those points.

150. Find the locus of the point \(z\) if \(\log_{\sqrt{3}}\frac{|z|^2 - |z| + 1}{2 + |z|} < 2.\)