This solution will guide you through solving a problem involving hyperbolas, focusing on key properties like foci, eccentricity, latus rectum, and the condition for perpendicular lines.

1. Understanding the Standard Hyperbola and its Foci

• Key Concept: For a hyperbola with the standard equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where the transverse axis lies along the x-axis, the coordinates of its foci are $(\pm ae, 0)$. Here, $a$ is the semi-transverse axis length, $b$ is the semi-conjugate axis length, and $e$ is the eccentricity. The eccentricity $e$ is related to $a$ and $b$