Conic Sections

Concept Map

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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The specific shape of a conic section, which can be a circle, ellipse, parabola, or hyperbola, depends on the ______ and ______ of the intersecting plane.

angle

position

Standard equation of a circle

(x-h)² + (y-k)² = r² where (h,k) is the center and r is the radius.

Circle's symmetry characteristic

Circle has perfect symmetry around its center (h,k), similar to both foci of an ellipse.

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Ellipses: The Geometry of Orbits

An ellipse is a closed curve resembling a flattened circle, with two focal points known as foci. It is created when a plane intersects a cone at an oblique angle that is not perpendicular to the cone's axis. The longest and shortest diameters of an ellipse are termed the major and minor axes, respectively. The general equation for an ellipse centered at (h,k) with semi-major axis a and semi-minor axis b is given by (x-h)²/a² + (y-k)²/b² = 1. The foci are situated along the major axis, symmetrically spaced from the center, and the ellipse's shape is determined by the eccentricity, which is less than 1 for all ellipses.

Parabolas: Curves of Quadratic Equations

A parabola is a mirror-symmetrical, open curve that is produced when a plane intersects a cone parallel to one of its generatrices—the slanted lines that define the cone's surface. Parabolas are unique among conic sections in having a single focus and a corresponding directrix, which is a line perpendicular to the axis of symmetry. The vertex of the parabola is the point where the curve turns and is the midpoint between the focus and the directrix. The standard form of a parabola's equation with vertex (h,k) is y = a(x-h)² + k for a vertical orientation, or x = a(y-k)² + h for a horizontal orientation, where a determines the width of the parabola.

Hyperbolas: Open Curves with Two Asymptotes

Hyperbolas are composed of two separate, unbounded branches that appear as mirror images of each other, opening in opposite directions. This conic section arises when a plane intersects both halves of a double cone, but not at the apex. The center of a hyperbola is the midpoint of the line segment joining its vertices—the points where each branch is closest to the other. The transverse axis connects the vertices, while the conjugate axis is perpendicular to it. The standard equation for a hyperbola centered at (h,k) with transverse axis length 2a and conjugate axis length 2b is (x-h)²/a² - (y-k)²/b² = 1 for a horizontal orientation, or (y-k)²/a² - (x-h)²/b² = 1 for a vertical orientation. Hyperbolas have two foci and two directrices, with the foci located on the transverse axis and the directrices parallel to the conjugate axis.

Analyzing and Graphing Conic Sections

Conic sections can be represented graphically on a Cartesian coordinate system using their respective standard equations. The general second-degree equation for conic sections is Ax² + Bxy + Cy² + Dx + Ey + F = 0, where A, B, C, D, E, and F are constants that determine the type and orientation of the conic. The eccentricity (e) is a parameter that describes the conic section's deviation from a circular shape: e = 0 for circles, 0 < e < 1 for ellipses, e = 1 for parabolas, and e > 1 for hyperbolas. Mastery of these equations and the associated concepts of focus, directrix, and eccentricity is crucial for accurately graphing conic sections and solving complex problems in geometry and calculus.

Conic Sections

Concept Map

Summary

Outline

Conic Sections

Definition and Types

Intersection of a plane and a right circular cone

Types of conic sections

Practical applications

Equations and Properties

Standard equations

Eccentricity

Focus and Directrix

Graphing and problem solving

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