In a complex number plane, consider the complex numbers $z$ such that $z^3$ is a real number. (1) Let $C$ be the figure formed by the set of complex numbers $z = x + iy$ satisfying the above condition. Since the arguments of the complex numbers $z$ satisfy $$\arg z = \frac{\pi}{\mathbf{M}}k \quad (k : \text{integer}),$$ figure $C$ consists of three straight lines represented in terms of $x$ and $y$ by the equations $$y = \mathbf{N}, \quad y = \sqrt{\mathbf{O}}\,x, \quad y = -\sqrt{\mathbf{O}}\,x.$$ (2) Suppose that on $C$ there exists only one complex number $z$ satisfying $|z - 1 - i| = r$. Then the value of $r$ is $$r = \frac{\sqrt{\mathbf{Q}} - \square\mathbf{R}}{\square}$$ and the value of $z$ is $$z = \frac{\mathbf{T} + \sqrt{\mathbf{U}}}{\square\mathbf{V}}(1 + \sqrt{\mathbf{W}}\,i).$$
In a complex number plane, consider the complex numbers $z$ such that $z^3$ is a real number.
(1) Let $C$ be the figure formed by the set of complex numbers $z = x + iy$ satisfying the above condition. Since the arguments of the complex numbers $z$ satisfy
$$\arg z = \frac{\pi}{\mathbf{M}}k \quad (k : \text{integer}),$$
figure $C$ consists of three straight lines represented in terms of $x$ and $y$ by the equations
$$y = \mathbf{N}, \quad y = \sqrt{\mathbf{O}}\,x, \quad y = -\sqrt{\mathbf{O}}\,x.$$
(2) Suppose that on $C$ there exists only one complex number $z$ satisfying $|z - 1 - i| = r$. Then the value of $r$ is
$$r = \frac{\sqrt{\mathbf{Q}} - \square\mathbf{R}}{\square}$$
and the value of $z$ is
$$z = \frac{\mathbf{T} + \sqrt{\mathbf{U}}}{\square\mathbf{V}}(1 + \sqrt{\mathbf{W}}\,i).$$