Let $x_n = a_n + b_n i$, where $a_n , b_n \in \mathbb{R}$, and $ n \in \mathbb{N} $, and suppose that the sequence ${x_n}$ is governed by the recurrence relation $x_{n+2} = x_n + i\,m x_{n+1}$, where $m \in \mathbb{R}$ . We investigate the conditions under which two subsequences $x_{2k}$ and $x_{2k+1}$ lie on perpendicular lines in the complex plane. Specifically, we express $x_1$ and $x_2$ as $x_1 = a_1 + b_1 i $ and $ x_2 = a_2 + b_2 i$, satisfying the condition $ a_1 a_2 + b_1 b_2 = 0 $.

We show that there exist two perpendicular lines $ l_1$ and $l_2$ passing through the origin in the complex plane, such that for all $k \in \mathbb{N} $, the sequence alternates between the lines: $ x_{2k} \in l_1 $ and $ x_{2k+1} \in l_2$. This provides a geometric interpretation of the recurrence relation. Furthermore, the ratios $ \frac{b_{2k}}{a_{2k}} = \frac{b_2}{a_2}$ and $\frac{b_{2k+1}}{a_{2k+1}} = \frac{b_1}{a_1} $remain constant for all $k \in \mathbb{N} $.