You are correct!
From our dictionary:
fat•u•ous
(fach'OO•us),
—adj.
1. foolish or inane, esp. in an unconscious, complacent manner; silly.
2. unreal; illusory.
(fach'OO•us),
—adj.
1. foolish or inane, esp. in an unconscious, complacent manner; silly.
2. unreal; illusory.
