Step 1: Set up the clinical picture.
The patient has recurrent hemoptysis (coughing blood), a normal chest X-ray, and purulent sputum. Purulent sputum with recurrent bleeding on a normal X-ray points toward a chronic airway disease that a plain film can miss, most often bronchiectasis.
Step 2: Recall the best test for suspected bronchiectasis.
High resolution CT (HRCT) of the chest gives thin, detailed slices of the lung that can show the dilated, thick-walled airways typical of bronchiectasis even when the chest X-ray looks normal. It is the most sensitive test for this diagnosis.
Step 3: Compare with spiral CT.
Spiral (helical) CT scans the chest continuously and is better suited to picking up pulmonary embolism or a vascular cause of bleeding, such as an active bleeding vessel before embolization. It is not the first test chosen when the picture points to a chronic airway problem rather than a vascular one.
Step 4: Compare with MRI.
MRI is not good at imaging air-filled lung tissue because it has poor spatial resolution for lung parenchyma, so it is not used to work up bronchiectasis or hemoptysis.
Step 5: Compare with bronchoscopy.
Bronchoscopy is very useful for finding the exact bleeding site, taking biopsies, or ruling out a tumor, but it is usually done after imaging has localized the disease, or when imaging is inconclusive, not as the very next step when the picture already suggests bronchiectasis.
Final Answer:
The next investigation of choice here is
\[ \boxed{\text{HRCT chest}} \]