Embodied Question Answering (EQA) combines visual scene understanding, goal-directed exploration, spatial and temporal reasoning under partial observability. A central challenge is to confine physical search to question-relevant subspaces while maintaining a compact, actionable memory of observations. Furthermore, for real-world deployment, fast inference time during exploration is crucial. We introduce FAST-EQA, a question-conditioned framework that (i) identifies likely visual targets, (ii) scores global regions of interest to guide navigation, and (iii) employs Chain-of-Thought (CoT) reasoning over visual memory to answer confidently. FAST-EQA maintains a bounded scene memory that stores a fixed-capacity set of region-target hypotheses and updates them online, enabling robust handling of both single and multi-target questions without unbounded growth. To expand coverage efficiently, a global exploration policy treats narrow openings and doors as high-value frontiers, complementing local target seeking with minimal computation. Together, these components focus the agent's attention, improve scene coverage, and improve answer reliability while running substantially faster than prior approaches. On HMEQA and EXPRESS-Bench, FAST-EQA achieves state-of-the-art performance, while performing competitively on OpenEQA and MT-HM3D.