\(A(a, 0)\) is a fixed point and \(\theta\) is a parameter such that \(0 < \theta < 2 \pi\). If \(\mathrm{P}(\mathrm{a} \cos \theta, \mathrm{a} \sin \theta)\) is a point on the circle \(\mathrm{x}^2+\mathrm{y}^2=\mathrm{a}^2\) and \(\mathrm{Q}(\mathrm{b} \sin \theta,-\mathrm{b} \cos \theta)\) is a point on the circle \(x^2+y^2=b^2\), then the locus of the centroid of the triangle APQ is

1. A a circle with centre at \(\left(\frac{\mathrm{a}}{3}, 0\right)\) and radius \(\left(\frac{\sqrt{\mathrm{a}^2+\mathrm{b}^2}}{3}\right)\)
2. B \(a\) circle with centre at \((a, 0)\) and radius \(\left(\frac{\sqrt{a^2+b^2}}{3}\right)\)
3. C a parabola with focus at \(\left(\frac{\mathrm{a}}{3}, 0\right)\)
4. D a parabola with focus at \((a, 0)\)