# 21. Complex Numbers Problems Part 3

1. 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 $$\frac{a^{-1} + b^{-1} + c^{-1} + d^{-1}}{a^{-1}b^{-1}c^{-1}d^{-1}}$$.

2. 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.

3. 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 $$arg\left( \frac{z_3}{z_2}\right) = arg\left( \frac{z_3 - z_1}{z_2 - z_1}\right)^2.$$

4. $$P$$ 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 $$z_2^2 cos2\theta = z_1z_3 cos^2\theta.$$

5. Find the equation in complex variables of all circles which are orthogonal to $$|z| = 1$$ and $$|z - 1| = 4.$$

6. 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 $$z - 3\sqrt{3}i < t^2.$$

7. 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.

8. 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

$x_1^2 + y_1^2 - x_1 = \frac{x_2^2 + y_2^2 + 2x_2 - 2y_2 +1}{(x_2 - 1)^2 + y_2^2}$
9. Simplify the following:

$\frac{(cos3\theta - isin3\theta)^6(sin\theta - icos\theta)^3}{(cos2\theta + isin2\theta)^5}$
10. Find all complex numbers such that $$z^2 + |z| = 0.$$

11. Solve the equation $$z^2 + z|z| + |z^2| = 0.$$

12. Solve the equation $$2z = |z| + 2i$$ in complex numbers.

13. If $$a > 0$$ and $$z|z| + az + 1 = 0$$, show that $$z$$ is a negative real number.

14. For every real number $$a > 0$$ find all complex numbers $$z$$ satisfying the equation $$z|z| + az + i = 0$$

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

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

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

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

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

20. If $$a_i, b_i \in R, i = 1, 2, 3, ..., n$$ show that

$\left(\sum_{n=1}^na_i\right)^2 + \left(\sum_{n=1}^nb_i\right)^2 \le \left(\sum_{n=1}^n\sqrt{a_i^2 + b_i^2}\right)^2$
21. 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 $$|z_1| = 2.$$

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

23. If $$z_1$$ and $$z_2$$ are complex numbers and $$u = \sqrt{z_1z_2},$$ prove that

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

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

26. If $$|z + 4| \le 3,$$ find the least and greatest value of $$|z + 1|.$$

27. Show that for any two non-zero complex numbers $$z_1, z_2$$

$(|z_1| + |z_2|)\left(\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right) \le 2|z_1 + z_2|$
28. 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 $$|a| \le 2, |b| = 1, argb = 2arga$$

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

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

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

32. 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

$\prod_{i = 1}^n(1 + \alpha_i^2) = (1 - a_2 + a_4 - ...)^2 + (a_1 - a_3 + ...)^2$
33. 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 $$z_1 + z_2 + z_3 = 0.$$

34. 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.

35. 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 $$z_1^2 + z_2^2 + z_3^2 = 3z_0^2.$$

36. Prove that the complex numbers $$z_1$$ and $$z_2$$ and the origin form an equilateral triangle if $$z_1^2 + z_2^2 - z_1z_2 = 0.$$

37. 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 $$a^2 = 3b.$$

38. 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 $$\alpha^2 = \beta.$$

39. If $$z_1, z_2$$ and $$z_3$$ are in A.P., then prove that they are collinear.

40. If $$z_1, z_2$$ and $$z_3$$ are collinear points in Argand plane then show that one of the following holds

$-z_1|z_2 - z_3| + z_2|z_3 - z_1| + z_3|z_1 - z_2| = 0 z_1|z_2 - z_3| - z_2|z_3 - z_1| + z_3|z_1 - z_2| = 0 z_1|z_2 - z_3| + z_2|z_3 - z_1| - z_3|z_1 - z_2| = 0$
41. Find the locus of point $$z$$ if $$\frac{z - i}{z + i}$$ is purely imaginary.

42. What region in the Argand plane is represented by the inequality $$1 < |z - 3 -4i| < 2.$$

43. Find the locus of point $$z$$ if $$|z - 1| + |z + 1| \le 4.$$

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

45. 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.

46. If $$|z^2 - 1| = |z|^2 + 1,$$ show that locus of $$z$$ is a straight line.

47. Find the locus of point $$z$$ if $$\frac{\pi}{3} \le arg~z \le \frac{3\pi}{2}.$$

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

49. 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.

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