A person stands at the middle point of a wooden ladder which starts slipping between a vertical wall and the floor of a room, while continuing to remain in a vertical plane. The path traced by a person standing at the middle point of the slipping ladder is:

Imagine the ladder is a line segment of length L. Let the base of the ladder on the floor be at point X and the top of the ladder against the wall be at point Y. In a coordinate system where the floor is the x axis and the wall is the y axis, the coordinates of the ends are X, 0 and 0, Y.

According to the Pythagorean theorem, X squared plus Y squared equals L squared.

The person is at the midpoint of the ladder. The coordinates of this midpoint are x equals X divided by 2 and y equals Y divided by 2. This means X equals 2x and Y equals 2y.

Substituting these into the first equation, we get 2x squared plus 2y squared equals L squared. This simplifies to 4x squared plus 4y squared equals L squared, or x squared plus y squared equals L squared divided by 4.

This is the standard equation of a circle centered at the origin with a radius of L divided by 2. Therefore, the path traced by the person at the midpoint is a circular path.