Person stands at the middle point of a wooden ladder which starts slipping between a vertical wall and the floor of a room Why is the path traced by a person standing at the middle of ladder circular?

Suppose the wall is represented by the y-axis and the ground as the x-axis. Let the position of the man be given by M = (x, y).

Then,

since M is the midpoint of the ladder, the base of the ladder is at A = (2x, 0).

also, the point where it touches the wall is B = (0, 2y).

Now, AB is the length of the ladder so AB = constant.

Using Pythagoras's theorem,

(2x - 0)2 + (0 - 2y)2 = c

(2x - 0)2 + (2y - 0)2 = c

This is the equation of [a quadrant of] a circle with centre (0, 0) and radius = sqrt(c).