Ellipses and Hyperbolas

This page covers the properties and equations of ellipses and hyperbolas.

Ellipses

An ellipse is defined as the locus of points where the sum of distances from two fixed points (foci) is constant.

The standard form of an ellipse centered at the origin is:

Formula: x²/a² + y²/b² = 1

Where a and b are the lengths of the semi-major and semi-minor axes, respectively.

Key concepts include:

• Eccentricity (e = c/a, where c² = a² - b²)
• Foci locations: F(±c, 0) for horizontal ellipse, F(0, ±c) for vertical ellipse

Vocabulary: The eccentricità ellisse (eccentricity of an ellipse) is a measure of how much it deviates from a circular shape.

Hyperbolas

A hyperbola is defined as the locus of points where the difference of distances from two fixed points (foci) is constant.

The standard form of a hyperbola centered at the origin is:

Formula: x²/a² - y²/b² = 1 (for horizontal hyperbola) Formula: y²/a² - x²/b² = 1 (for vertical hyperbola)

Key concepts include:

• Eccentricity (e = c/a, where c² = a² + b²)
• Asymptotes: y = ±(b/a)x
• Equilateral hyperbola: a = b, equation becomes x² - y² = a²

Highlight: The equazione parabola: esercizi (parabola equation exercises) and ellisse formule (ellipse formulas) are crucial for understanding these conic sections.

The page concludes with information on the rational function and translated hyperbolas.

Advanced Concepts

This final section covers advanced topics including homographic functions and translated conics.

Definition: A homographic function is expressed as y = (ax + b)/(cx + d), where c ≠ 0 and ad - bc ≠ 0.

Example: Translated hyperbolas follow the form: (x - xc)²/a² - (y - yc)²/b² = 1

Highlight: The asymptotes of hyperbolas provide important information about the curve's behavior at infinity.

Vocabulary: The center and asymptotes are essential elements in understanding translated conics.