Problem 1: Find the center, vertices, foci, and the equations of the asymptotes for the hyperbola given by $\frac{x^2}{16} - \frac{y^2}{9} = 1$.

Problem 2: Convert the general equation $9x^2 - 4y^2 - 36x - 8y - 4 = 0$ into standard form and determine the coordinates of its center.

Problem 3: Find the standard equation of a hyperbola with vertices at $(\pm 5, 0)$ and foci at $(\pm 13, 0)$.

Case Study 1: Two LORAN (Long Range Navigation) stations, $A$ and $B$, are located $400\text{ km}$ apart. A ship receives a navigational radio signal from station $A$ exactly $1000\text{ microseconds}$ ($1000 \mu\text{s}$) before it receives the synchronized signal from station $B$. Assuming radio signals travel at $300\text{ m/}\mu\text{s}$, determine the equation of the hyperbolic path the ship lies on, placing the midpoint between the stations at the origin.

Problem 5: Calculate the exact length of the latus rectum and the eccentricity for the hyperbola defined by $\frac{y^2}{36} - \frac{x^2}{64} = 1$.