Introduction to Hyperbolas and Their Conjugates

This problem requires a solid understanding of the fundamental properties of a hyperbola and its conjugate. Specifically, we need to recall the formulas for their eccentricity and the length of their latus rectum.

1. Standard Hyperbola ($H$): A standard hyperbola with its transverse axis along the x-axis has the equation: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad (\text{where } a > 0, b > 0)$

   • Eccentricity ($e$): This value measures how "open" the hyperbola is. For a hyperbola, $e > 1$. The relationship between $a, b,$ and $e$ is given by: $e^2 = 1 + \frac{b^2}{a^2}$
   • Length of Latus Rectum ($l$): The latus rectum is a chord passing through a focus and perpendicular to the transverse axis. Its length is: $l = \frac{2b^2}{a}$

2. Conjugate Hyperbola ($H'$): The conjugate hyperbola to $H$ has its transverse axis along the y-axis. Its equation is: $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$ When defining its properties, the semi-transverse axis becomes $b$ and the semi-conjugate axis becomes $a$. This effectively swaps the roles of $a$ and $b$ in the formulas for eccentricity and latus rectum compared to the standard hyperbola.

   • Eccentricity ($e'$): For the conjugate hyperbola, the relationship between $a, b,$ and $e'$ is: $(e')^2 = 1 + \frac{a^2}{b^2}$ Self-check: Notice how $a$ and $b$ have swapped positions in the fraction compared to the standard hyperbola's eccentricity formula.
   • Length of Latus Rectum ($l'$): For the conjugate hyperbola, its length of latus rectum is: $l' = \frac{2a^2}{b}$ Self-check: Similarly, $a$ and $b$ have swapped roles in this formula.

Now, let's apply these definitions to solve the given problem step-by-