A rod of length 12cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3cm from the end in contact with the x-axis.

The problem can be represented as follows:
Given AB = 12 cm, AP = 3 cm, PB = 12 - 3 = 9cm.
In $\Delta PBQ, \cos \theta = \large\frac{PQ}{PB} $$= \large\frac{PQ}{9} In \Delta PRA, \sin \theta = \large\frac{PR}{PA}$$ = \large\frac{PR}{3}$
We know that $\sin^2 \theta + \cos^2\theta = 1 \rightarrow \large(\frac{PQ}{9})^2 $$+ \large(\frac{PR}{3})^2$$=1$
This gives us the equation of the locus of point P as $\large\frac{x^2}{81}$$+\large\frac{y^2}{9}$$=1$