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

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(b>a)$ is an ellipse with eccentricity $\frac{1}{\sqrt{2}}$. If the angle of intersection between the ellipse and parabola $y^2=4 a x$ is $\theta$, then the coordinates of the point $\frac{2 \theta}{3}$ on the ellipse is

The number of common tangents that can drawn to the curves $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $x^2+y^2=16$ is

Let $A(\alpha, 4,7)$ and $B(3, \beta, 8)$ be two points in space. If $Y Z$ plane and $Z X$-plane respectively divide the line segment joining the points $A$ and $B$ in the ratio $2: 3$ and $4: 5$, then the point $C$ which divides $A B$ in the ratio $\alpha: \beta$ externally is