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Conic sections are curves like circles, ellipses, parabolas, and hyperbolas formed by intersecting a plane with a cone. Each has unique properties and equations, crucial in mathematics and practical applications such as orbital mechanics and architecture. Understanding their geometry involves mastering concepts like focus, directrix, and eccentricity.

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## Definition and Types

### Intersection of a plane and a right circular cone

Conic sections are formed by the intersection of a plane and a right circular cone

### Types of conic sections

Circles

Circles are formed when a plane intersects a cone at a right angle and does not pass through the apex

Ellipses

Ellipses are formed when a plane intersects a cone at an oblique angle and has two focal points

Parabolas

Parabolas are formed when a plane intersects a cone parallel to one of its generatrices and have a single focus and directrix

Hyperbolas

Hyperbolas are formed when a plane intersects both halves of a double cone and have two foci and two directrices

### Practical applications

Conic sections have practical applications in fields such as orbital mechanics, optics, and architecture

## Equations and Properties

### Standard equations

Each type of conic section has a unique standard equation that defines its geometry

### Eccentricity

The eccentricity of a conic section describes its deviation from a circular shape

### Focus and Directrix

Conic sections have foci and directrices that determine their shape and orientation

### Graphing and problem solving

Understanding the equations and concepts of focus, directrix, and eccentricity is crucial for accurately graphing conic sections and solving complex problems in geometry and calculus

Algorino

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