Consider complex numbers $z$ such that $$z \bar { z } - ( 1 - 2 i ) z - ( 1 + 2 i ) \bar { z } \leqq 15 .$$ (1) On a complex number plane, the figure represented by inequality (1) is the interior and circumference of the circle having the center $\mathbf{L} + \mathbf{M} i$ and the radius $\mathbf{NO}$. (2) Let us consider all complex numbers $z$ which are on the straight line $$( 1 - i ) z - ( 1 + i ) \bar { z } = 2 i$$ and satisfy the inequality (1). Of those, let $z _ { 1 }$ be the $z$ such that $| z |$ is maximized and $z _ { 2 }$ be the $z$ such that $| z |$ is minimized. Then we have $$z _ { 1 } = \sqrt { \mathbf { P Q } } + \mathbf{Q} + ( \sqrt { \mathbf { S T } } + \mathbf { U } ) i ,$$ $$z _ { 2 } = - \frac { \mathbf { U } } { \mathbf { V } } + \frac{\mathbf{W}}{\mathbf{P}} i .$$
Consider complex numbers $z$ such that
$$z \bar { z } - ( 1 - 2 i ) z - ( 1 + 2 i ) \bar { z } \leqq 15 .$$
(1) On a complex number plane, the figure represented by inequality (1) is the interior and circumference of the circle having the center $\mathbf{L} + \mathbf{M} i$ and the radius $\mathbf{NO}$.
(2) Let us consider all complex numbers $z$ which are on the straight line
$$( 1 - i ) z - ( 1 + i ) \bar { z } = 2 i$$
and satisfy the inequality (1). Of those, let $z _ { 1 }$ be the $z$ such that $| z |$ is maximized and $z _ { 2 }$ be the $z$ such that $| z |$ is minimized. Then we have
$$z _ { 1 } = \sqrt { \mathbf { P Q } } + \mathbf{Q} + ( \sqrt { \mathbf { S T } } + \mathbf { U } ) i ,$$
$$z _ { 2 } = - \frac { \mathbf { U } } { \mathbf { V } } + \frac{\mathbf{W}}{\mathbf{P}} i .$$