When the origin is shifted to the point $(h, k)$ by translating the coordinates axes, the equation $S \equiv 2 x^2-x y+y^2+2 x+3 y+1=0$ is changed to $S \equiv a x^2+2 h x y+b y^2-3=0$. Again by rotating the coordinate axes about the new origin through the angle $\theta$ in the positive direction, $S^{\prime}=0$ is changed to $A x^2+B y^2+C=0$. Then, $h+k+\tan 2 \theta=$

In an ellipse, the distance from one of the foci to its corresponding end of the major axis is $4-\sqrt{7}$ and the distance from same focus to one end of the minor axis is 4 . Then, the cosine of the angle subtended by the line segment joining its foci at one end of its minor axis is

If the equations $x=1+2 \cos \theta, y=2+\sin \theta, 0 \leq \theta<2 \pi$ represent an ellipse, then the point of intersection of the normal drawn at $P\left(\frac{\pi}{4}\right)$ to this ellipse and its major axis is

Let $A=(2,0)$ and $B=(0,-2)$. Let $P$ be any point such that the sum of the distance of $P$ from $A$ and $B$ is 4 . Then, the equation of the locus of the point $P$ is