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

Let $P$ be the point to which origin has to be shifted by the translation of axes, so as to remove the first degree terms from the equation $3 x^2+y^2-6 x+4 y+4=0$. If the origin is shifted to $P$ by the translation of axes, then the transformed equation of $2 x^2+3 x y-5 y^2+2 x-23 y-24=0$ is