Conic SectionsParabola Ellipse

Hyperbola

ZAHIDADEPARTMENT OF MATHEMATICS

UNIVERSITY OF HAIL

Conic Sections

Contents• General Equation of a Conic Section Second Degree Equation • Parabola: Definition-- Algebraic and Geometric Examples• Ellipse Definition-- Algebraic and Geometric Examples• Hyperbola Definition-- Algebraic and Geometric Examples

General Equation of a Conic Section

022 FEyDxCyBxyAx

•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)e

PL

PF

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.

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⇒

pxy 42

Examples: Parabola p = 1 p > 0

p = -1 p < 0pxy 42

Parabolapyx 42

• Axis of symmetry y-axis

• Vertex (0,0)

• Focus (0,p)

• Directrix y = -p

• p>0 Parabola opens up

p<0 Parabola opens down

Example: Parabola

p > 0 p < 0

pyx 42

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).

Ellipse Algebraic Definition

12

2

2

2

b

y

a

x

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

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)

1916

22

yx

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)

1819

22

yx

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.

Hyperbola Algebraic Definition

12

2

2

2

b

y

a

x

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.

Hyperbola Algebraic Definition

12

2

2

2

b

y

a

x1

2

2

2

2

b

y

a

x

The standard form of a hyperbola with a 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

The standard form of a hyperbola with center at (0,0) and a horizontal axis is

•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

xa

by

•)

xa

by

Example: Hyperbola

12

2

2

2

b

y

a

x1

2

2

2

2

b

y

a

x

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