Question 2
To explore space, scientists often observe distant objects such as comets, asteroids and other astronomical phenomena. For this purpose they design and launch observation satellites into orbit around Earth. Most satellites do not move in perfect circles but have elliptical orbits, with Earth at one of the two foci. As a satellite moves in an elliptical orbit, its distance from the object being observed continually changes. Usually, the satellite's sensors receive the best signal when this distance is smallest.
Consider a satellite, regarded as a point, orbiting Earth along an ellipse. In three-dimensional Cartesian coordinates $Oxyz$, with each axis unit equal to $6400$ km, suppose the satellite moves in the coordinate plane $(Oxy)$ along $x^2+3y^2=17$. It must observe an object, also regarded as a point, moving in space. Research indicates that the best observation occurs when the object reaches $A(2;16/\sqrt3;8)$. Determine the coordinates of the point $C$ on the satellite's elliptical orbit for which the distance from $C$ to $A$ is smallest.
Consider a satellite, regarded as a point, orbiting Earth along an ellipse. In three-dimensional Cartesian coordinates $Oxyz$, with each axis unit equal to $6400$ km, suppose the satellite moves in the coordinate plane $(Oxy)$ along $x^2+3y^2=17$. It must observe an object, also regarded as a point, moving in space. Research indicates that the best observation occurs when the object reaches $A(2;16/\sqrt3;8)$. Determine the coordinates of the point $C$ on the satellite's elliptical orbit for which the distance from $C$ to $A$ is smallest.