A rod of length 9 cm moves with its ends always touching the co-ordinate axes. Show that the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis, is an ellipse.

Let AB be the rod of length 9 cm with end A touching the y-axis and end B touching the x-axis.
                                        
P is a point on the rod, such that BP = 3 cm
∴ AP = 9 - 3 = 6 cm.
We have to find the locus of point P
Let the co-ordinate of point P be 
From P, draw PL and PM perpendicular to x-axis and y-axis respectively.
∴                           
Now, from similar triangles PLB and AOB,
                               
Also, from similar triangles AMP and AOB,
                                 
Using Pythagoras theorem, we get
                           
              
                
Hence, the equation of the locus of point 
which is the equation of the ellipse in standard form