8/7/2019 [LEC]1.2 Conic Sections

1/28

Conic Sections

8/7/2019 [LEC]1.2 Conic Sections

2/28

CalculusDe La Salle University

Parabolas

Ellipses

Hyperbolas

Topics:

8/7/2019 [LEC]1.2 Conic Sections

3/28

8/7/2019 [LEC]1.2 Conic Sections

4/28

CalculusDe La Salle University

Given a quadratic equation

the graph is either a parabola, an ellipse ora hyperbola. For special cases it can be an

empty set, a point or intersecting lines.

GRAPH OF A QUADRATIC EQUATION

Conic Section

022

FEyDxCyBxyAx

8/7/2019 [LEC]1.2 Conic Sections

5/28

CalculusDe La Salle University

Aparabola is the set of all points in a plane

equidistant from a fixed point and a fixed

line. The fixed point is called the focus whilethe fixed line is called the directrix.

PARABOLA

Parabolas

8/7/2019 [LEC]1.2 Conic Sections

6/28

CalculusDe La Salle University

axis

focus

parabola

latus rectum

vertex

directrix

Parabolas

8/7/2019 [LEC]1.2 Conic Sections

7/28

CalculusDe La Salle University

An equation of the parabola having its focus

at the point(0,p) and having as its directrix

the line y = p is

EQUATION OF A PARABOLA

Parabolas

pyx 42

8/7/2019 [LEC]1.2 Conic Sections

8/28

CalculusDe La Salle University

An equation of the parabola having its focus

at the point(p,0) and having as its directrix

the line x = p is

EQUATION OF A PARABOLA

Parabolas

pxy 42

8/7/2019 [LEC]1.2 Conic Sections

9/28

CalculusDe La Salle University

Example:

Sketch the graph of the given parabola.

Indicate its vertex, axis, focus, equation of

directrix, and endpoints of latus rectum.1.

2.

3. Focus at (0,-1/2); directrix:

4. Vertex at origin, opening upward thru (-4,-2)

Parabolas

yx 82

yy 122

012 y

8/7/2019 [LEC]1.2 Conic Sections

10/28

CalculusDe La Salle University

An equation of the parabola with a vertical

axis having its vertex at the point(h,k) and p

as the directed distance from the vertex tothe focus is

PARABOLA IN STANDARD FORM

Parabolas

)(4)(2 kyphx

8/7/2019 [LEC]1.2 Conic Sections

11/28

CalculusDe La Salle University

An equation of the parabola with a

horizontal axis having its vertex at the point

(h,k) and p as the directed distance from thevertex to the focus is

PARABOLA IN STANDARD FORM

Parabolas

)(4)(2 hxpky

8/7/2019 [LEC]1.2 Conic Sections

12/28

CalculusDe La Salle University

Example:

Sketch the graph of the given parabola.

Indicate its vertex, axis, focus, equation of

directrix, and endpoints of latus rectum.1.

2.

3. Focus at (1,-4); directrix: x - axis

4. Vertex at (-3,2), opening rightward thru (-2,3)

Parabolas

62

41

xxy

yyx 6122

8/7/2019 [LEC]1.2 Conic Sections

13/28

CalculusDe La Salle University

An ellipse is the set of points in a plane the

sum of whose distance from two fixed points

is a constant. Each fixed points is called afocus.

ELLIPSE

Ellipses

8/7/2019 [LEC]1.2 Conic Sections

14/28

CalculusDe La Salle University

minor axis

major axis

focus

vertices center

Ellipses

8/7/2019 [LEC]1.2 Conic Sections

15/28

CalculusDe La Salle University

If 2a is the constant sum referred to in the

definition of an ellipse and the foci are at

(c,0) and (-c,0), then the equation of theellipse is

where b2 = a2 c2.

EQUATION OF AN ELLIPSE

Ellipses

12

2

2

2

b

y

a

x

8/7/2019 [LEC]1.2 Conic Sections

16/28

CalculusDe La Salle University

If an ellipse has its foci at (0,c) and (0,-c),

then the equation of the ellipse is

where b2 = a2 c2.

EQUATION OF AN ELLIPSE

Ellipses

12

2

2

2

b

x

a

y

8/7/2019 [LEC]1.2 Conic Sections

17/28

CalculusDe La Salle University

Example:

Sketch the graph of the given ellipse. Indicate

its vertices, center, foci and endpoints of the

minor axis.

1.

2.

11625

22

yx

1169144

22

yx

Ellipses

8/7/2019 [LEC]1.2 Conic Sections

18/28

CalculusDe La Salle University

If the center of an ellipse is at (h,k) and the

distance between the two vertices is 2a,

then the equation of an ellipse is of the form

(horizontal major axis)

(vertical major axis)

ELLIPSE IN STANDARD FORM

Ellipses

1)()(

2

2

2

2

b

ky

a

hx

1)()(

2

2

2

2

b

hx

a

ky

8/7/2019 [LEC]1.2 Conic Sections

19/28

CalculusDe La Salle University

Example:Sketch the graph of the given ellipse. Indicate

its vertices, center, foci and endpoints of the

major and minor axis.1.

2.

3.

4.

01119128150162522

yxyx

0361503625922

yxyx

0115542496 22 yxyx

010554249622

yxyx

Ellipses

8/7/2019 [LEC]1.2 Conic Sections

20/28

CalculusDe La Salle University

Example:Find the equation of an ellipse in general form

having the given properties then sketch its

graph.1. Vertices at (-5/2,0) and (5/2,0) and a focus at

(3/2,0).

2. Foci at (-1,-1) and (-1,7) and a minor axis oflength 8 units.

Ellipses

8/7/2019 [LEC]1.2 Conic Sections

21/28

CalculusDe La Salle University

A hyperbola is the set of points in a plane,

the absolute value of the difference of

whose distances from two fixed points is aconstant. Each of the fixed points is called a

focus.

HYPERBOLA

Hyperbolas

8/7/2019 [LEC]1.2 Conic Sections

22/28

CalculusDe La Salle University

Hyperbolas

asymptoteauxiliary rectangle

conjugate axis

transverse axis

center

foci

vertices

8/7/2019 [LEC]1.2 Conic Sections

23/28

CalculusDe La Salle University

If 2a is the constant referred to in the

definition of an hyperbola and the foci are

at (c,0) and (-c,0), then the equation of thehyperbola is

where b2 = c2 a2.

EQUATION OF A HYPERBOLA

Hyperbolas

12

2

2

2

b

y

a

x

8/7/2019 [LEC]1.2 Conic Sections

24/28

CalculusDe La Salle University

If a hyperbola has its foci at (0,c) and (0,-c),

then the equation of the hyperbola is

where b2 = c2 a2.

EQUATION OF A HYPERBOLA

Hyperbolas

12

2

2

2

b

x

a

y

8/7/2019 [LEC]1.2 Conic Sections

25/28

CalculusDe La Salle University

Example:Sketch the graph of the given hyperbola.

Indicate its vertices, center, foci auxiliary

rectangle and equation of the asymptotes.

1.

2.

1916

22

yx

194

22

xy

Hyperbolas

8/7/2019 [LEC]1.2 Conic Sections

26/28

CalculusDe La Salle University

If the center of a hyperbola is at (h,k) and

the distance between the two vertices is 2a,

then its equation is of the form

(horizontal transverse

axis)(vertical transverse

axis)

HYPERBOLA IN STANDARD FORM

Hyperbolas

1)()(

2

2

2

2

b

ky

a

hx

1)()(

2

2

2

2

b

hx

a

ky

8/7/2019 [LEC]1.2 Conic Sections

27/28

CalculusDe La Salle University

Example:Sketch the graph of the given hyperbola.

Indicate its vertices, center, foci auxiliary

rectangle and equation of the asymptotes.1.

2.

02916184922

yxyx

05754249622

yxyx

Hyperbolas

8/7/2019 [LEC]1.2 Conic Sections

28/28

CalculusD L S ll U i it

Hyperbolas

Example:Find the equation of a hyperbola in general

form having the given properties then sketch

its graph.1. Vertices at (-5,-3) and (-5,-1) and the endpoints

of the conjugate axis at (-7,-2) and (-3,-2).

2. Center at (3,-5), a vertex at (7,-5) and a focusat (8,-5)