Locus of a General Point on Falling Ladder

In continuation to my previous post on locus of mid-point of a falling ladder , let us now explore the locus of a point lying anywhere on the falling ladder.

Example : A 6-foot ladder is placed vertically against a wall, and then the foot of the ladder is moved outward until the ladder lies flat on the floor with one end touching the wall. What is the locus of the point which divides the ladder in the ratio 2 : 3 as it slides?

Solution: Let P be the point which divides the slider AB (A (a,0) being the feet and B (0,b) rest on the wall) .O is the intersection point of floor and wall.As the feet of the ladder slides,the point P whose coordinates are(3a/5,2b/5) travels on an elliptical path given by x²/9+y²/4=(6/5)².

In the following applet, press the play button at the lower left corner to see the various positions of the ladder as it is dragged and the movement of point P of the ladder. You can also select the ‘Show Locus’ button to see the full locus and its equation.