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**Conic Sections Parabola Ellipse Hyperbola**

ZAHIDA DEPARTMENT OF MATHEMATICS UNIVERSITY OF HAIL

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Conic Sections

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**Contents General Equation of a Conic Section Second Degree Equation**

Parabola: Definition-- Algebraic and Geometric Examples Ellipse Hyperbola

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**General Equation of a Conic Section**

A conic section is the locus of a second degree equation in two variables x and y. Let F be a fixed point called focus, L a fixed line called directrix in a plane, and e be a fixed positive number called eccentricity. The set of all points P in the plane such that is a conic section. (e=1 Parabola, e<1 ellipse, e>1 hyperbola)

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**What is Parabola Geometrically?**

A parabola is the set of all points in a plane such that each point in the set is equidistant from a line called the directrix and a fixed point called the focus.

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**Parabola Algebraic Definition**

The Standard Form of a parabola that opens to the right or left and has a vertex (0,0) Axis of symmetry x-axis Vertex (0,0) Focus (p, 0) Directrix x = -p p>0 ⇒ Parabola opens towards right p<0 ⇒ Parabola opens towards left

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Examples: Parabola p = 1 p > 0 p = -1 p < 0

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**Parabola Axis of symmetry y-axis Vertex (0,0) Focus (0,p)**

Directrix y = -p p>0 Parabola opens up p<0 Parabola opens down

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Example: Parabola p > 0 p < 0

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**What is Ellipse Geometrically?**

The set of all points in the plane, the sum of whose distances from two fixed points, called the foci, is a constant and is equal to major axis (length).

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**Ellipse Algebraic Definition**

The standard form of an ellipse with a center at (0,0) and a horizontal axis (that is major axis is x-axis)

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Example: Ellipse The ellipse with a center at (0,0) and a horizontal axis has the following characteristics Major axis is along x-axis Vertices ( ± a,0) (ends of major axis) Co-Vertices (0,±b) (ends of minor axis) Foci ( ± c,0) (on major axis)

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Example: Ellipse The ellipse with a center at (0,0) and a vertical axis has the following characteristics Major Axis Along y-axis Vertices (0, ± b) (ends of major axis) Co-Vertices (±a,0) (ends of minor axis) Foci (0, ± c) (on major axis)

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**What is hyperbola Geometrically?**

The set of all points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.

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**Hyperbola Algebraic Definition**

The standard form of a hyperbola with a center at (0,0) and a horizontal axis is Transverse axis has length 2a. Conjugate axis has length 2b. If x2 is the term with positive sign, the transverse axis of the hyperbola is horizontal If y2 is the term with positive sign, the transverse axis of the hyperbola is vertical.

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**Hyperbola Algebraic Definition**

The standard form of a hyperbola with a center at (0,0) and a horizontal axis is The standard form of a hyperbola with center at (0,0) and a horizontal axis is Transverse axis is along x-axis and its length is 2a. Vertices ( ± a,0) (ends of transverse axis) Foci (±c,0) (on transverse axis) Asymptotes are lines Transverse axis is along y-axis and its length is 2b. Vertices (0, ± b) (ends of transverse axis) Foci (0, ± c) (on transverse axis) Asymptotes are )

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Example: Hyperbola

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THANKS

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