A circle of radius \(a\) rolls round the outside of a closed oval curve whose total perimeter is \(s\) and whose area is \(S\). Show that the locus of the centre of the circle is an oval curve of perimeter \(s+2\pi a\) enclosing an area \(S+as+\pi a^2\). \newline If the circle rolls on the inside of the oval curve and is sufficiently small to be always entirely within it, show that the locus of the centre is another oval of perimeter \(s-2\pi a\) enclosing an area \(S-as+\pi a^2\).