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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
THANKS