If $P(\theta)$ and $Q\left(\frac{\pi}{2}+\theta\right)$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the locus of mid-point of $P Q$ is $\frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}=1$, then $\frac{a+b}{\alpha+\beta}=$

Let $S$ be the focus of the hyperbola $x^2-2 y^2=1$ lying on the positive $X$-axis. Let $P(-1,1)$ be a given point. Then, the area of the triangle formed by the line $P S$ with the coordinate axes is (in sq. units)

If $P\left(\frac{\pi}{6}\right)$ is a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, S, S$ are its foci and $S P+S P=2 | S P-S P$|, then $e=$

Let $A=(1,2,0), B=(2,0,-1), C=(0,-2,3)$ and $D=(-1,2,-3)$ be four points in the space. Let $G_1$ be the centroid of $\triangle A B C$ and $G_2$ be the centroid of tetrahedron $A B C D$. If $P$ divides, $G_1 G_2$ in the ratio $4: 3$ internally, then $P=$