# 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 $$\frac{z_2z_3}{z_1}.$$