A ladder of length L stands against a vertical wall. It is being slipped down against the wall. Find the locus of

(a) a point which divides ladder 2:3 internally from floor side.

(b) mid-point of ladder.

a) Say the point on the ladder is C(x,y). The ends of the ladder are A(a,0) B(0,b). Here, a and b are not constants as the ladder is slipping down.
As C divides AB in 2:3,
x = 3a/5
y = 2b/5
As length of ladder is constant,
a^2 + b^2 = L^2
(x/3)^2 + (y/2)^2 = (L/5)^2
This is the locus. It is an ellipse.
b) Mid point of ladder :
D(x,y) is the midpoint.
x = a/2