Why are PAPI lights installed 300m from the threshold? What are the conditions that define the position of the PAPI lights from threshold?
US DoT Order JO 6850.2B contains installation criteria for visual guidance lighting systems. The following is taken from that order.
For siting PAPI along the runway, in case an (ILS) electronic glide slope is present, the lights are located such that the PAPI visual approach path coincides with the one produced electronically as much as possible.
In case of runways without an electronic glide slope, the following criteria are considered for locating the PAPI:
- Threshold Crossing Height
The TCH is the height of the lowest on-course signal at a point directly above the intersection of the runway centerline and the threshold. The minimum allowable TCH varies according to the height group of aircraft that use the runway, ... The PAPI approach path must provide the proper TCH for the most demanding height group that uses the runway.
- Glide Path Angle
The visual glide path angle is the center of the on-course zone, and shall normally be 3 degrees when measured from the horizontal.
- Obstacle Clearance Surface
The PAPI obstacle clearance surface is established to provide the pilot with a minimum clearance over obstacles during approach. The PAPI must be positioned and aimed so that no obstacles penetrate this surface.
Image from US DoT JO 6850.2B Visual Guidance Lighting Systems
Basically, in case ILS is there, align the PAPI with that; else locate the PAPI such that it satisfies TCH and OCS.
(Image Source: WikiPedia)
In this picture, you see that the PAPI is placed right next or near to the touchdown zone.
(Image Source: Aviation.SE)
Since you want guidance for this area, placing them near the threshold would actually make you aim too close to the threshold and increase the possibility of touching down too short of the runway.