Consider an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ in the $xy$-plane. Here, $a$ and $b$ are constants satisfying $a > b > 0$. Answer the following questions.
Find the equation of the tangent line at a point $(X, Y)$ on the ellipse in the first quadrant.
The tangent line obtained in Question I. 1 intersects the $x$- and $y$-axes. Find the coordinates $(X, Y)$ at the tangent point that minimizes the length of the segment connecting the two intersects and obtain the minimum length of the segment.
Consider a region bounded by the segment obtained in Question I. 2 and the $x$- and $y$-axes, and let $C_{1}$ be a cone formed by rotating the region around the $x$-axis. Next, let $C_{2}$ be a cone with the maximum volume while having the same surface area (including a base area) as the cone $C_{1}$. Find $\frac{S_{2}}{S_{1}}$, where $S_{1}$ and $S_{2}$ are the base areas of the cones $C_{1}$ and $C_{2}$, respectively.
Consider an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ in the $xy$-plane. Here, $a$ and $b$ are constants satisfying $a > b > 0$. Answer the following questions.
\begin{enumerate}
\item Find the equation of the tangent line at a point $(X, Y)$ on the ellipse in the first quadrant.
\item The tangent line obtained in Question I. 1 intersects the $x$- and $y$-axes. Find the coordinates $(X, Y)$ at the tangent point that minimizes the length of the segment connecting the two intersects and obtain the minimum length of the segment.
\item Consider a region bounded by the segment obtained in Question I. 2 and the $x$- and $y$-axes, and let $C_{1}$ be a cone formed by rotating the region around the $x$-axis. Next, let $C_{2}$ be a cone with the maximum volume while having the same surface area (including a base area) as the cone $C_{1}$. Find $\frac{S_{2}}{S_{1}}$, where $S_{1}$ and $S_{2}$ are the base areas of the cones $C_{1}$ and $C_{2}$, respectively.
\end{enumerate}