Can you photograph a shooting star?

Yes, and it's easier than you think. You need a camera with manual settings (a phone with night mode works too), a tripod or stable surface, and patience. Wide lens (14-24mm on full frame). Aperture wide open (f/2.8 or lower). ISO 1600-3200. Exposure 15-30 seconds. Focus on infinity. Point at a dark patch of sky. Take hundreds of shots. Most will be empty. A few will catch a streak. The trick is volume. Meteor showers give you better odds. The Geminids are great for photography — slower meteors, brighter, often colorful. You're not just taking a picture. You're catching a 4-billion-year-old rock's last seconds.