A rod PQ of length 2a slides with its ends on the axes. The locus of the circumcentre of $∆$OPQ is 
 

$$\begin{gathered} \mathrm{\Delta OPQ} \mathrm{is} \mathrm{a} \mathrm{right} \mathrm{triangle} \\ \mathrm{Circumcentre} \mathrm{of} \mathrm{right} \mathrm{triangle} \mathrm{is} \mathrm{mid} \mathrm{point} \mathrm{oh} \mathrm{Hypotenuse} \mathrm{PQ}=2\mathrm{a} \\ \mathrm{let} \mathrm{M} \mathrm{is} \mathrm{the} \mathrm{mid} \mathrm{point} \mathrm{of} \mathrm{PQ} \\ \mathrm{M}=(\frac{\mathrm{a}+0}{2},\frac{0+\mathrm{b}}{2})\Rightarrow \mathrm{M}(\mathrm{h},\mathrm{k})=(\frac{\mathrm{a}}{2},\frac{\mathrm{b}}{2}) \\ \Rightarrow \mathrm{h}=\frac{\mathrm{a}}{2} , \mathrm{k}=\frac{\mathrm{b}}{2}\iff \mathrm{a}=2\mathrm{h}, \mathrm{b} = 2\mathrm{k} \\ {\mathrm{PQ}}^2={\mathrm{OP}}^2+{\mathrm{OQ}}^2 \\ 4{\mathrm{a}}^2={\mathrm{b}}^2+{\mathrm{a}}^2 \\ \Rightarrow 4{\mathrm{a}}^2=4{\mathrm{k}}^2+4{\mathrm{h}}^{2^{ }} \\ \mathrm{So} \mathrm{the} \mathrm{locus} \mathrm{of} \mathrm{circumcentre} \mathrm{of} \mathrm{\Delta OPQ} \mathrm{is} {\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{a}}^2 \end{gathered}$$