# 16. Complex Numbers Problems Part 4¶

If \(O\) be the center of the circle circumscribing the equilateral \(\triangle ABC\) and its radius be unity and \(A\) lies on the \(x\)-axis. Find the complex numbers represented by \(B\) and \(C.\)

\(ABCD\) is a rhombus. Its diagonals \(AC\) and \(BD\) intersect at a point \(M\) and satisfy \(BD = 2AC.\) If the points \(D\) and \(M\) represent the complex numbers \(1 + i\) and \(2 - i\) respectively, then find the complex number represented by \(A\).

If \(z_1, z_2, z_3\) and \(z_4\) are the vertices of a square taken in anticlockwise order, prove that \(z_3 = -iz_1 + (1 + i)z_2\) and \(z_4 = (1 - i)z_1 + iz_2.\)

Let \(z_1, z_2\) and \(z_3\) are the vertices of an equilateral triangle in the circle \(|z| = 2.\) If \(z_1 = 1 + i\sqrt{3},\) then find \(z_2\) and \(z_3.\)

If \(a\) and \(b\) are real numbers between 0 and 1 such that the points \(z_1 = a + i, z_2 = 1 + bi,\) and \(z_3 = 0\) form an equilateral triangle then find \(a\) and \(b.\)

Let \(ABCD\) be a square described in the anticlockwise sense in the Argand plane. If \(A\) represents \((3 + 5i)\) and the center of the square represents \(\left(\frac{7}{2} + \frac{5}{2}i\right).\) Find the numbers represented by \(B, C\) and \(D\).

Find the vertices of a regular polygon of \(n\) sides, if its center is located at origin and one of its vertices is \(z_1.\)

Prove that the points \(a(cos\alpha + i sin\alpha), b(cos\beta + i sin\beta)\) and \(c(cos\gamma + i sin\gamma)\) in the Argand plane are collinear if \(bcsin(\beta - \gamma) + casin(\gamma - \alpha) + absin(\alpha -\beta) = 0.\)

\(A\) represents the number \(6i, B\) the number \(3\) and \(P\) the complex number \(z\). If \(P\) moves such that \(PA : PB = 2 : 1,\) show that \(z\overline{z} = (4 + 2i)z + (4 - 2i)\overline{z}.\) Also, show that the locus of P is a circle, find its radius and center.

Show that if the points \(z_1, z_2, z_3\) and \(z_4\) taken in order are concyclic then the expression \(\frac{(z_3 - z_1)(z_4 - z_2)}{(z_3 - z_2)(z_4 -z_1)}\) is purely real.

Let \(z_1, z_2, z_3\) and \(z_4\) be the vertices of a quadrilateral. Prove that quadrilateral is cyclic if \(z_1z_2 + z_3z_4 = 0\) and \(z_1 + z_2 = 0.\)

Show that the triangles whose vertices are \(z_1, z_2, z_3\) and \(z_1', z_2', z_3'\) are similar if

\[\begin{split}\begin{vmatrix}z_1 & z_1' & 1\\z_2 & z_2' & 1\\z_3 & z_3' & 1\end{vmatrix} = 0\end{split}\]If \(a, b, c\) and \(u, v, \omega\) are the complex numbers representing two triangles such that \(c = (1 - r)a + rb\) and \(\omega = (i - r)u + rv,\) where \(r\) is a complex number, prove that the two triangles are similar.

Find the equation of perpendicular bisector of the line segment joining points \(z_1\) and \(z_2.\)

Find the equation of the circle having the line segment joining \(z_1\) and \(z_2\) as diameter.

If \(\left|\frac{z - z_1}{z - z_2}\right| = c, c \ne~0,\) then show that the locus of \(z\) is a circle.

If \(|z| = 1,\) find the locus of the point \(\frac{2}{z}.\)

If for any two complex numbers \(z_1\) and \(z_2, |z_1 + z_2| = |z_1| + |z_2|\) prove that \(arg(z_1) - arg(z_2) = 2n\pi.\)

Find the complex number \(z\) the least in absolute value which satisfies the condition \(|z - 2 + 2i| = 1.\)

Find the point in the first quadrant, on the curve \(|z - 5i| = 3\) whose argument is minimum.

Find the set of points of the coordinate plane which satisfy the inequality

\[log_\frac{1}{2}\left(\frac{|z - 1| + 4}{3|z - 1| - 2}\right) > 1\]Find the set of all points on the \(xy\)-plane whose coordinates satisfy the following conditions: the number \(z^2 + z + 1\) is real and positive.

Find the real values of the parameter \(a\) for which at least one complex number \(z\) satisfies the equality \(|z - az| = a + 4\) and the inequality \(|z - 2| < 1.\)

Find the real values of parameter \(t\) for which at least one complex number \(z\) satisfies the equality \(|z + \sqrt{2}| = t^2 - 3t +2\) and the inequality \(|z + t\sqrt{2}| < t^2.\)

Among the complex numbers which satisfy the condition \(|z - 15i| \le 15,\) find the number having least positive argument.

Find \(z\) for which \(|z - 5i| \le 1\) and \(arg(z)\) is minimum.

Find the real value of \(a\) for which there is at least one complex number satisfying \(|z + 4i| = \sqrt{a^2 - 12a + 28}\) and \(|z - 4\sqrt{3}| < a.\)

Find the set of points belonging to the coordinate plane \(xy,\) for which the real part of complex number \((1 + i)z^2\) is positive.

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

Show that all the roots of the equation \(a_1z^3 + a_2z^2 + a_3z + a_4 = 3,\) where \(|a_i| \le 1, i= 1, 2, 3, 4\) lie outside the circle with center as origin and radius \(\frac{2}{3}.\)

Given that \(\sum_{i = 1}^nb_i = 0\) and \(\sum_{i = 1}^nb_iz_i = 0\) where \(b_i\)’s are non-zero real numbers, no three of \(z_i\)’s form a straight line. Prove that \(z_i\)’s are concyclic if \(b_1b_2|z_2 - z_2|^2 = b_3b_4|z_3 - z_4|^2.\)

A cubic equation \(f(x) = 0\) has one real root \(\alpha\) and two complex roots \(\beta + i\gamma.\) Points \(A, B\) and \(C\) represent these roots respectively. Show that the roots of the derived equation \(f'(x) = 0\) are complex if \(A\) falls inside one of the two equilateral triangles described on base \(BC\).

Prove that reflection of \(\overline{a}z + a\overline{z} = 0\) in the real axis is \(\overline{a}\overline{z} + az = 0.\)

If \(\alpha, \beta, \gamma, \delta\) are four complex numbers such that \(\frac{\gamma}{\delta}\) is real and \(\alpha\delta - \beta\gamma \ne 0\) then prove that \(z = \frac{\alpha + \beta t}{\gamma + \delta t}, t \in \mathbf{R}\) represents a straight line.

If \(\omega, \omega^2\) are cube roots of unity, then prove that

- \((3 + 3\omega + 5\omega^2)^2 - (2 + 6\omega + 2\omega^2) = 0\)
- \((2 - \omega)(2 - \omega^2)(2 -\omega^{10})(2 - \omega^{11}) = 49\)
- \((1 - \omega)(1 - \omega^2)(1 - \omega^4)(1 - \omega^5) = 9\)
- \((1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 = 32\)
- \(1 + \omega^n + \omega^{2n} = 3\) where \(n > 0, n \in \mathbf{I}\) and is a multiple of 3.
- \(1 + \omega^n + \omega^{2n} = 0\) where \(n > 0, n \in \mathbf{I}\) and is not a multiple of 3.

Resolve into linear factors \(a^2 + b^2 + c^2 -ab -bc -ca\)

Prove that

\[\sqrt{-1 - \sqrt{-1 - \sqrt{-1 - ... to \infty}}} = \omega \text{ or } \omega^2\]If \(x = a + b, y = a\omega + b\omega^2, z = a\omega^2 + b\omega,\) prove that \(x^3 + y^3 + z^3 = 3(a^3 + b^3)\) and \(xyz = a^3 + b^3.\)

Resolve into linear factors:

- \(a^2 -ab + b^2\)
- \(a^2 + ab + b^2\)
- \(a^3 + b^3\)
- \(a^3 - b^3\)
- \(a^3 + b^3 +c^3 -3abc\)

Show that \(x^{3p} + x^{3q+1} + x^{3r+2},\) where \(p, q, r\) are positive integers is divisible by \(x^2 + x + 1.\)

Show that \(x^{3p} + x^{3q+1} + x^{3r+2} + x^{4s+3},\) where \(p, q, r, s\) are positive integers is divisible by \(x^3 + x^2 + x + 1.\)

If \(p = a + b + c, q = a + b\omega + c\omega^2, r = a + b\omega^2 + c\omega,\) where \(\omega\) is a cube root of unity, prove that \(p^3 + q^3 + r^3 - 3pqr = 27abc.\)

If \(\omega\) be an imaginary root of unity, prove that \((a + b\omega + c\omega^2)^3 + (a + b\omega^2 + c\omega)^3 = (2a - b -c)(2b -a -c)(2c -a -b).\)

If \(ax + cy + bz = X, cx + by + az = Y, bx + ay + cz = Z\) show that

- \((a^2 + b^2 + c^2 - ab - bc - ca)(x^2 + y^2 + z^2 - xy - yz -zx) = X^2 + Y^2 + Z^2 - XY -YZ -ZY\)
- \((a^3 + b^3 - c^3 - 3abc)(x^3 + y^3 + z^3 -3xyz) = X^3 + Y^3 + Z^3 - 3XYZ\)

Prove that

\[\left(\frac{cos\theta + isin\theta}{sin\theta + icos\theta}\right)^4 = cos8\theta + isin8\theta\]If \(z^2 -2zcos\theta +1 =0,\) show that \(z^2 + z^{-2} = 2cos2\theta\)

If \(x_r = cos\frac{\pi}{2^r} + isin\frac{\pi}{2^r},\) prove that \(x_1x_2x_3 ... to \infty = -1.\)

Prove that \((1 + i)^n + (1 - i)^n = 2^{\frac{n}{2} + 1}cos\frac{n\pi}{4}.\)

Show that the value of

\[\sum_{k = 1}^6\left(sin\frac{2\pi k}{7} -icos\frac{2\pi k}{7}\right)\]is \(i.\)

Show that

\[e^{2mi cot^{-1}p}\left(\frac{pi + 1}{pi - 1}\right)^m = 1\]