Consider a rod of length l aligned along the y axis. A slight disturbance caused the bottom of the rod to slip along the positive x axis. Find the path and its equation followed by the point at a distance r from the lower end of the rod.

the path followed by the point will be an ellipse.

imagine the rod as AB with A slipping along the Y axis and B slipping along the X axis. let B be initially located at the origin and later on it slips along the positive direction of X axis. the center of mass of the rod will always fall along the Yaxis as net force on the system is purely in the vertical direction(its own weight, assuming no walls are present to support the rod during its motion). call the required point as P and let at any position the rod make an angle * with the horizontal(negative X axis).

the co ordinates of the point P at that time will be ((L/2 - r)cos*, rsin*). call this point as (X, Y). now eliminate * and the equation becomes X²/(L/2 - r)²  +  Y²/r²  = 1.

THIS IS THE PATH FOLLOWED BY THE POINT.

the path will be an ellipse