This document contains mathematical problems on analytic geometry, specifically focusing on concepts such as triangles, ellipses, and hyperbolas. Here’s an outline of the types of questions included:

1. Triangles and Lines: Calculating equations for the sides of a triangle given certain line equations and points.
2. Angles Between Lines: Determining the smallest angle between two given lines.
3. Right Triangles: Finding the hypotenuse and other sides using given line equations and points.
4. Distance to a Point: Finding the equation of a line equidistant from a specific point and axis.
5. Ellipses: Working with ellipses defined by eccentricities, focal points, and directrices, and calculating distances, axes, and center points.
6. Identifying and Characterizing Conic Sections: Given certain equations, identifying whether they represent ellipses and finding the centers, axes, and eccentricities.
7. Hyperbolas: Finding focal points, axes, asymptotes, and directrices for hyperbolas, along with their eccentricities.

Would you like assistance with a specific problem from this set, or would you prefer a step-by-step explanation of any of the concepts involved?

Follow-up Questions:

1. Would you like an explanation of how to find the equation of a line passing through a given point?
2. Are you interested in the formula for calculating the angle between two lines?
3. Would you like to learn how to find the equation of an ellipse given its foci and eccentricity?
4. Do you need help with the steps for identifying and analyzing conic sections from their equations?
5. Would you like assistance in understanding how to derive the asymptotes of a hyperbola?

Tip: When solving for eccentricity in ellipses and hyperbolas, remember that the value can help determine the type of conic section and its specific properties.