Parabola Families and Special Cases

This page delves into more advanced topics related to parabolas, including degenerate parabolas and families of parabolas. It begins by discussing degenerate parabolas, which occur under special conditions.

Definition: Degenerate parabolas are special cases where the parabola reduces to a line or a point.

The main focus of this page is on fascio di parabole, or families of parabolas. Various methods for finding the equation of a family of parabolas are presented.

Example: To find the equation of a fascio di parabole passante per due punti, use the formula (y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁).

The page covers different scenarios for families of parabolas:

1. Parabolas passing through two given points
2. Parabolas passing through two distant points
3. Parabolas tangent to a line at a given point

Highlight: For a fascio di parabole passante per un punto and tangent to a line, use the equation y = ax² + bx + c + k(x - x₀)(y - mx₀ - q), where (x₀, y₀) is the tangent point and y = mx + q is the equation of the tangent line.

These advanced techniques are crucial for solving complex problems involving parabolas and their properties.

Vocabulary: Fascio di parabole - A family or set of parabolas sharing certain properties

The page provides a solid foundation for tackling fasci di parabole esercizi svolti and understanding the relationships between different parabolas in a family.