19. Complex Numbers Problems Part 2

  1. If equation \(z^2+\alpha z+\beta = 0\) has a real root then prove that \((\alpha\overline{\beta}-\beta\overline{\alpha})(\overline{\alpha}- \alpha)=(\beta-\overline{\beta})^2\)

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

  3. If \(iz^3+z^2-z+i=0,\) then show that \(|z|=1\)

  4. If \(\alpha\) and \(\beta\) are any two complex numbers then show that \(|\alpha+\sqrt{\alpha^2-\beta^2}| +|\alpha-\sqrt{\alpha^2-\beta^2}| = |\alpha+\beta|+|\alpha-\beta|\)

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

  6. Prove that \(\left|\frac{z_1-z_2}{1-\overline{z_1z_2}}\right|<1\) if \(|z_1|<1, |z_2|<1\)

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

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

  9. If \(z\le 1, |w|\le 1,\) show that \(|z - w|^2\le (|z| - |w|)^2 + (Arg(z) - Arg(w))^2\)

  10. If \(z\) is any non-zero complex number, show that

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

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

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

  13. Let \(z_1, z_2\) be any two complex numbers and \(a,b\) be two real numbers such that \(a^2+b^2 \ne 0.\) Prove that

    \[|z_1|2 + |z_2|^2 - |z_1^2 + z_2^2| \le 2\frac{|az_1+bz_2|^2}{a^2+b^2} \le |z_1|^2 + |z_2|^2 + |z_1^2 + z_2^2| \]
  14. If \(b+ic=(1+a)z\) and \(a^2+b^2+c^2=1,\) prove that \(\frac{a+ib}{1+c}=\frac{1+iz}{1-iz},\) where \(a,b,c\) are real numbers and \(z\) is a real number.

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

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

  17. If \(z_1, z_2, z_3\) be the vertices of an equilateral triangle, show that

    \[\frac{1}{z_1-z_2} + \frac{1}{z_2-z_3} + \frac{1}{z_3-z_1}=0 \text{ or } z_1^2+z_2^2+z_3^2 = z_1z_2+z_2z_3+z_3z_1 \]
  18. If \(z_1^2+z_2^2-2z_1z_2cos\theta,\) show that the points \(z_1,z_2\) and the origin are the vertices of an isosceles triangle.

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

  20. Prove that the area of triangle formed by three complex numbers \(z_1, z_2, z_3\) is

    \[\left|\sum\frac{(z_2-z_3)|z_1|^2}{4iz_1}\right| \]
  21. If \(n>1\) then show that the roots of the equation \(z^n=(z+1)^n\) are collinear.

  22. If \(A, B, C, \text{ and } D\) are four complex number then show that \(AD.BC\le BD.CA + CD.AB\)

  23. If \(a,b\in R \text{ and } a,b\ne 0\) then show that the equation of line joining these \(a\) and \(ib\) is

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

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

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

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

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

  29. Let \(z_1, z_2, z_3\) be three complex numbers and \(a, b ,c\) be real numbers not all zero such that \(a + b + c=0\) and \(az_1 + bz_2 + cz_3 = 0,\) then show that \(z_1, z_2, z_3\) are collinear.

  30. If \(z_1 + z_2 + ... +z_n = 0,\) 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.

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

    \[z_1 + z_2 + ... + z_n \ne 0 \text{ and } \frac{1}{z_1} + \frac{1}{z_2} + ... + \frac{1}{z_2} \ne 0 \]
  32. The points \(z_1 = 9+ 12i\) and \(z_2 = 6 - 8i\) are given on a complex plane. Find the equation of the bisector of the angle formed by the vector representing \(z_1\) and \(z_2.\)

  33. If the vertices of a triangle \(ABC\) are \(z_1, z_2, z_3,\) then show that the orthocenter of the \(\triangle ABC\) is

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

    where \(z\) is orthocenter and \(a, b, c\) are sides.

  34. If the vertices of a triangle \(ABC\) are \(z_1, z_2, z_3,\) then show that the circumcenter of the \(\triangle ABC\) is

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

    where \(z\) is circumcenter.

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

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

  37. Points \(z_1\) and \(z_2\) are adjacent points of a regular polygon with \(n\) sides. If \(z_3\) is adjacent vertex to \(z_2\) where \(z_2 \ne z_1\) then find \(z_3.\)

  38. If \(z_1, z_2\) and \(z_3\) are sides of an equilateral triangle and \(z_0\) is the centroid then prove that \(z_1^2 + z_2^2 + z_3^2 = 3z_0^2.\)

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

  40. If \(A_1, A_2, ..., A_n\) be the vertices of a regular polygon of \(n\) sides in a circle of radius unity. Find the values of \(|A_1A_2|^2 + |A_1A_3|^2 + ... + |A_1A_n|^2\) and \(|A_1A_2||A_1A_3| ... |A_1A_n|\)

  41. If \(|z|=2\) the show that points representing the complex numbers \(-1 + 5z\) lie on a circle.

  1. If \(z-6-8i \le 4\) then find the greatest and least value of \(z.\)

  2. If \(z-25i \le 15\) then find the least positive value of \(argz.\)

  3. Show that the equation \(|z - z_1|^2 + |z-z_2|^2 = k\) where \(k \in R\) will represent a circle if \(k \ge \frac{1}{2}|z_1 - z_2|^2.\)

  4. If \(|z-1| = 1,\) prove that \(\frac{z-2}{z} = i tan(rag z).\)

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

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

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

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

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

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