Unless you had registered the software with Adobe and could therefore access your Adobe account at this site to check on your registration yourself, you are Severely Out of Luck.
Even if you had the serial number, you can only install an Adobe application on two computers that will not be used simultaneously. A third installation would violate the terms of the License (EULA) and would not be activated.
Photoshop 9 was actually Photoshop CS2, and it hasn't been sold by Adobe for some 7 or 8 years.
If you mean Adobe Photoshop Elements 9, that is not "Photoshop", and there is a dedicated Elements forum, although the answer you would get there would be identical to this post,
Incidentally, these are user to user forums. You are not addressing Adobe here.