The circumcenter of the equilateral triangle having the three points \(\theta_1, \theta_2, \theta_3\) lying on the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) as its vertices is \((r, s)\). Then the average of \(\cos \left(\theta_1-\theta_2\right)\), \(\cos \left(\theta_2-\theta_3\right)\) and \(\cos \left(\theta_3-\theta_1\right)\) is

1. A \(\frac{1}{2}\left[\frac{3 r^2}{a^2}+\frac{3 s^2}{b^2}-1\right]\)
2. B \(\frac{3}{2}\left[\frac{r^2}{a^2}+\frac{s^2}{b^2}\right]\)
3. C \(\frac{1}{3}\left[\frac{r^2}{a^2}+\frac{s^2}{b^2}\right]\)
4. D \(\frac{1}{3}\left[\frac{r^2}{a^2}+\frac{s^2}{b^2}+\frac{r s}{a b}\right]\)