Parabolas

This page focuses on the properties and equations of parabolas.

Definition and Equation

A parabola is defined as the locus of points equidistant from a fixed point (focus) and a fixed line (directrix).

The general equation of a parabola with vertex at (xᵥ, yᵥ) is given as:

Formula: y - yᵥ = a(x - xᵥ)² + b(x - xᵥ) + c

For a parabola with vertex at the origin and axis along the y-axis, the equation simplifies to:

Formula: y = ax²

Key Elements

The page discusses important elements of a parabola:

• Vertex
• Focus
• Directrix
• Axis of symmetry
• Latus rectum

Highlight: The distance from the vertex to the focus is equal to the distance from the vertex to the directrix.

Tangent Line

The slope of the tangent line to a parabola at a point (x₀, y₀) is given by:

Formula: m = 2ax₀ + b (for a parabola with axis parallel to y-axis)

Example: For a parabola y = ax² + bx + c, the tangent line at (x₀, y₀) has the equation: y - y₀ = (2ax₀ + b)(x - x₀)

The page also mentions the formula direttrice parabola (directrix formula) and the asse di simmetria parabola (axis of symmetry).

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.