If you would like to authorize someone to pick up an x-ray on your behalf, here is a sample template you can use to write an authorization letter to pick up an x-ray.
Subject: Authorization to Pick Up X-Ray
I had recently undergone an X-Ray of [Mention X-Ray details] at your hospital on [Date]. Since I am not well, I cannot get my X-Ray reports by visiting the hospital. Therefore, I am authorizing [Name] to collect the X-Ray reports from your hospital on my behalf.
I request you to kindly consider this authorization letter and hand over my X-Ray reports to him/her without fail.
For your reference, here are my details.
Date of birth:
In case you have any doubt regarding this authorization letter, kindly reach me at [Email/contact number]