It happened at about 5.15pm on Dawnay Road in Bradford on Saturday, November 16.
A red Honda motorbike and a black Skoda Octavia taxi crashed at the junction with Nene street.
The rider of the motorcycle, a man in his 30s, and his pillion passenger, an 18-year-old man, were both seriously injured in the collision.
The 18-year-old man remains in hospital.
Police are appealing for witnesses or anyone who may have dashcam footage of the collision to contact them.
Anyone with information should contact RPU via the West Yorkshire Police online chat service or by calling 101, quoting log 1306 of 16/11.