The crash happened at about 10.30am on Saturday, April 13 on Askwith Moor Road at the junction with Moor Lane, near Weston, Otley.
A cyclist travelling from Moor Lane and a Ducati motorcycle travelling towards the A59 crashed at the junction.
Both riders fell from their bikes as a result.
A 60-year-old man suffered serious injuries.
He has been taken to the Leeds General Infirmary.
The cyclist left the scene before the emergency services attended.
North Yorkshire Police are appealing for the cyclist to contact them.
Police are also appealing to any witnesses of the crash to contact them by calling 101, select option 2, and asking for TC Steve James of Harrogate Roads Policing Group.