You are working as an apprentice drafting technician at an engineering and architecture firm. A client is reviewing your CAD layout for a new corporate pavilion that features structural pillars designed in the shape of a hyperbola. In the layout, the client notices that the pillars form two separate, disconnected, mirror-image curves rather than a single continuous boundary. The client asks you: 'What is the technical term for each of these two individual curves, why does a hyperbola naturally split into two disconnected pieces instead of a single continuous loop, and how do these curves behave relative to the center and their guiding guidelines as they extend outward?'

Write a professional response (1-2 paragraphs) explaining this to the client. In your response, make sure you:

1. Identify the specific mathematical term for each of the two separate, mirror-image curves that make up a hyperbola.
2. Recall and describe the geometric property (using the concept of distances to two foci) that causes the hyperbola to split into two separate, disconnected pieces.
3. Describe how each curve behaves relative to the center of the hyperbola and its asymptotes as it extends outward.