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.

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.

Linear Equations and Circles

This page covers fundamental concepts of linear equations and circles in analytical geometry.

Linear Equations

The page begins with the explicit and implicit forms of linear equations. It introduces the formula retta passante per due punti (equation of a line passing through two points):

Formula: y - y₁ = $y₂ - y₁$/$x₂ - x₁$ * $x - x₁$

The coefficiente angolare retta passante per due punti (slope of a line passing through two points) is also provided:

Formula: m = $y₂ - y₁$/$x₂ - x₁$

Other important concepts include:

• Midpoint formula
• Distance between two points
• Distance from a point to a line

Highlight: The page emphasizes the relationship between parallel lines (same slope) and perpendicular lines (slopes are negative reciprocals of each other).

Circles

The section on circles introduces the general equation of a circle:

Formula: $x - x₀$² + $y - y₀$² = r²

Where (x₀, y₀) is the center and r is the radius.

The implicit form of the circle equation is also presented:

Formula: x² + y² + ax + by + c = 0

The page concludes with information on the family of circles and degenerate circles.

Vocabulary: A degenerate circle is a circle with a radius of zero, which reduces to a point.