The foot of the ladder RS in the following figure is slipping away from the wall RO.

Then the point P(a fixed point on the ladder) lies on

Let the point P be $$\left(\alpha\ ,B\right)$$

Let OS =a, OR=b and RS = c and $$\angle\ OSR=\theta\ $$ , PS = z

=> a=ccos$$\theta\ $$ and b = c sin$$\theta\ $$

$$\alpha\ =a\ -\ z\ \cos\theta\ $$ and $$\beta\ \ =z\ \sin\theta\ $$

$$\alpha\ \ =\ c\ \cos\theta\ \ -z\ \cos\theta\ $$

=> $$\cos\ \theta\ \ =\ \frac{\alpha}{c-z}$$ and $$\sin\theta\ \ \ =\ \frac{\beta\ }{z}$$

$$\ \frac{\alpha\ ^2}{\left(c-z\right)^2}+\ \frac{\beta\ ^2}{z^2}=1$$ which is an ellipse.