1.3 Conic Sections

1.3 Conic Sections

Def. A line lying entirely on the cone is called a generator of the cone, and all generators of a cone pass

through its vertex .

1.3 Conic Sections

Def. A conic section is the intersection of a plane and a right circular cone with two nappes.

Types of Conic Sections

A degenerate conic is either a point, a line or two intersecting lines.

Degenerate Conic Sections

Plane figures that can be obtained by the intersection of a double cone with a plane passing through the apex. These include

a point, a line, and intersecting lines.

A conic section is a curve obtained by intersecting a double-cone (two cones with the same axis and sharing a vertex) with a plane. There are four types of non-degenerate conic sections: circles, ellipses, parabolas and hyperbolas.

A non-degenerate conic is either a parabola, an ellipse or a hyperbola

Non-degenerate Conics If the cutting plane is parallel to one and

only one generator, the curve of intersection is called a parabola.

Non-degenerate ConicsIf the cutting plane is not parallel to any

generator; that is, it cuts all generators, the curve of intersection is called an ellipse

Non-degenerate Conics If the cutting plane is parallel to two

generators, the curve of intersection is a hyperbola.

This is a screenshot from the movie Sex, Lies and Videotape (1989). The image shows a projection of two circles onto a plane. The projection of any circle onto a plane forms a conic section. In this case, it is a hyperbola.

Flashlight Conic SectionsThe light emitted from a flashlight with a circular lens is projected as a cone from the light bulb.  You can create Flashlight Conic Sections by projecting the light at the wall, allowing the wall to be the plane and the light from the flashlight being the cone.  Try the following shapes:

Thm 1.3.1(Non-degenerate conic)A non-degenerate conic is a set of points

on the plane such that the ratio of the undirected distance of P from a fixed point (called focus) to the undirected distance of P from a fixed line not through the fixed point (called the directrix) is a constant.

Non-degenerate Conics

The constant ratio mentioned in Theorem 1.3.1 is called the eccentricity of the conic, which we will denote by e.

QPeFP =

Let P be a point on a conic with focus at F and let Q be the projection of P on the directrix. Then from Theorem 1.3.1,

Non-degenerate Conics

Def. The line through a focus and perpendicular to a directrix of a conic is called the principal axis of the conic. A point of intersection of the conic and its principal axis is called a vertex of the conic.

Thm 1.3.2 (Non-degenerate Conics)

Given the eccentricity e of a conic, the conic is a parabola if ; an ellipse if ; a hyperbola if .

1=e10 << e

1>e

1.4 The Parabola

Def. A parabola is the set of all points on the plane which are equidistant from a fixed point and a fixed line.

The Parabola

( ) pxypxQPFP +=+−⇒= 22

( ) ( ) 222 pxypx +=+−

( ) ( )[ ] ( ) ( )[ ]pxpxpxpxy −++−−+=⇒ 2

( ) ( ) pxxpy 4222 ==⇒pxy 42 =⇒

( ) ( ) 222 pxpxy −−+=⇒

Squaring both sides of the last equation, we get

Standard Equation of a Parabola

The equation is called the standard equation of a parabola with vertex at the origin, focus at , the line given by

as directrix, and the x-axis as its principal axis.

pxy 42 =

( )0,ppx −=

Parabola

If pxy 42 = is the equation of a parabola, then a. the parabola opens to right if ; 0>pb. the parabola opens to the left if .0<p

Standard Equation of a Parabola

The equation is called the standard equation of a parabola with vertex at the origin, focus at , the line given by

as directrix and the y-axis as its principal axis.

( )p,0py −=

pyx 42 =

ParabolaIf is the equation of a parabola, then

a. the parabola opens upward if ; 0>pb. the parabola opens downward if .0<p

pyx 42 =

Latus Rectum of the Parabola

Def. The line segment joining two (2) points on a parabola which passes through the focus of the parabola and perpendicular to its principal axis is called the latus rectumof the parabola.

Remark

The length of the latus rectum of a parabola given by or is

.p4pyx 42 =pxy 42 =

ExampleFor each of the given equation of a parabola, find the

vertexfocusprincipal axisdirectrixendpoints of the latus rectum

Draw a sketch of the parabola.

yx 82 −=

xy 162 =a. b.

Solution for (b) 2−=p

( ) ( ) ( )2,42,42,4 −−−=−± and

vertex

( ) ( )2,0,0 −=p

endpoints of the latus rectum are

yx 82 −=

principal axis axisy −

( )0,0

focus

parabola opens downward

directrix 2=⇒−= ypy

p2 units to the left and right of the focus,

Sketch of the graph of yx 82 −=

2=y

( )2,4 −( ) 2,4 −−( )2,0 −F

Solution for (a) 4=p

( ) ( ) ( ) ( )8,48,48,42,4 −=±=± and p

vertex

( ) ( )0,40, =p

endpoints of the latus rectum are

xy 162 =

principal axis axisx −

( )0,0

focus

parabola opens righttheto

directrix 4−=⇒−= xpx

Sketch of the graph of xy 162 =

( )0,4F

4−=x

( ) 8,4

( )8,4 −

V

General Equation of the Parabola

If the vertex of a parabola is at the point and the principal axis of the parabola is

the line , the standard equation of the parabola is

(As before, the parabola opens to the right or to the left depending on the sign of p.)

( )k,h

ky =

( ) ( )hxpky −=− 42

General Equation of the Parabola

If the vertex of a parabola is at the point and the principal axis of the parabola is

the line , the standard equation of the parabola is

(As before, the parabola opens upward or downward depending on the sign of p.)

( )k,h

hx =

( ) ( )2 4x h p y k− = −

ExampleFor each of the given equation of a

parabola, find the following: the coordinates of the vertex and the focus, the principal axis, an equation of the directrix, and the coordinates of the endpoints of the latus rectum. Draw a sketch of the parabola:

a. b.( ) ( )131 2 −=+ yx ( ) ( )244 2 −−=+ xy

Solution for (a)43=p

−

=

±−

415,

25

415,

21

415,21 and p

vertex

( )

−=+−

415,13,1 p

endpoints of the latus rectum are

principal axis 1−=x

( )3,1−

focus

parabola opens upward

directrix 493 =⇒−=−= yppky

p2 units to the left and right of the focus,

( ) ( )331 2 −=+ yx ( )3434 −

= y

Sketch of the graph of ( ) ( )331 2 −=+ yx

( )3,1−

1−=x

Solution for (a) 1−=p

( ) ( ) ( )6,12,124,1 −−=±− and p

vertex

( ) ( )4,14,2 −=−+ p

endpoints of the latus rectum are

principal axis 4−=y

( )4,2 −

focus

parabola opens lefttheto

directrix 32 =⇒−=−= xpphx

p2 units upward and downward

( ) ( )244 2 −−=+ xy

Sketch of the graph of

( )3,1−

4−=y

( ) ( )244 2 −−=+ xy

Sources of Figures/Pictures

http://math2.org/math/algebra/conics.htmhttp://www.mathacademy.com/pr/prime/articles/conics/index.asp

http://en.wikipedia.org/wiki/Conic_sectionhttp://xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html

http://www.clausentech.com/lchs/dclausen/algebra2/conic_sections.htm

End of 1.4

1.5 The Ellipse

An ellipse is the set of all points on the plane, the sum of whose distances from two fixed points is a constant.

The fixed points referred to are called the foci of the ellipse.

Suppose the foci have coordinates(c,0) and (-c,0), 2a is the constant sum and if P(x,y) is any point in the ellipse then

a2 PF PF ___

2

___

1 =+

( ) ( ) a2ycxycx 2222 =++++−→

( ) ( ) 2222 ycxa2ycx ++−=+−→

( ) ( ) ( ) 2222222 ycxycxa4a4ycx +++++−=+−→

F1 (c,0)F2(-c,0)

P(x,y)

( ) cx4a4ycxa4 222 +=++→

( ) 22242222 xccxa2ayccx2xa ++=+++→

( ) cxaycxa 222 +=++→

( ) ( ) )1(22222222 caayaxca −=+−→

( ) 2222222 ccx2xycxa4a4ccx2x +++++−=+−→

In the triangle ,FPF 21∆

FF PF PF _____

21

___

2

___

1 >+

c2a2 >→ca >→

0ca 22 >−→

.1by

ax

2

2

2

2=+

F1 (c,0)F2(-c,0)

P(x,y)

.bca 222 =−Let .bayaxb 222222 =+Then

22baDividing (1) by becomes

( ) ( )22222222 caayaxca −=+−→

.1by

ax

2

2

2

2=+Consider

The x-intercepts of the graph are a and –a.

The y-intercepts of the graph are b and –b.

The principal axis is the x-axis.

The points (a,0) and (-a,0) are the vertices of the

ellipse.

(a,0)(-a,0)

(0,b)

(0,-b)

(a,0)(-a,0)

(0,b)

(0,-b)

The intersection of the

major axis and the minor

axis of the ellipse is called

its center.

The line segment joining the points (0,b) and (0,-b) is called the

minor axis of the ellipse.

The line segment joining the vertices is called the major axis of the ellipse.

Since , then . Hence, the major axis of the ellipse is always longer than its

minor axis.

222 bca =− ba >

Since , then . This ratio is the eccentricity of the ellipse while the

directrices of the ellipse are at the lines

ca > 1ac <

.ca

eax

2±=±=

• center at the origin

• x-axis is the principal axis• vertices at (a,0) and (-a,0)• foci at (c,0) and (-c,0) with c2 = a2 - b2

The equation 1by

ax

2

2

2

2=+

is the standard equation of the ellipse with

where ba >

• endpoints of the minor axis at (0,b) and (0,-b)

• equation of directrices given by ca

eax

2±=±=

(a,0)

(-a,0)

(0,b)

(0,-b)

Interchanging the roles of x and y, the equation will become

1bx

ay

2

2

2

2=+

How would this affect the properties of the ellipse?

• center at the origin

• y-axis is the principal axis• vertices at (0,a) and (0,-a)

• foci at (0,c) and (0,-c) with c2 = a2 - b2

1bx

ay

2

2

2

2=+The equation

is the standard equation of the ellipse with

where ba >

• endpoints of the minor axis at (b,0) and (-b,0)

• equation of directrices given by ca

eay

2±=±=

(0,-a)

(0,a)

(b,0)(-b,0)

Example. Given the ellipse with equation

determine the principal axis, vertices, endpoints of the minor axis, lengths of the major and minor axes, foci, eccentricity

and equations of directrices. Draw also a sketch of the ellipse.

14y

9x 22

=+

SOLUTION

From the standard equation,

.2band3a ==

Solving for the value of c,

222 bac −=49c2 −=

5c =

14y

9x 22

=+

T h e r e f o r e ,

• principal axis: • vertices:

• endpoints of the minor axis

• foci

• eccentricity

• equations of the directrices given by

14y

9x 22

=+x-axis

( ) ( )0,30,3 −and( ) ( )2,02,0 −and

( ) ( )0,5and0,5 −• length of the major axis• length of the minor axis

64

35=

ac

559

59x ±=±=

(3,0)(-3,0)

(0,2)

(0,-2)

( )0,5( )0,5−

559x =

559x −=

14y

9x 22

=+

Example. Given the ellipse with equation

determine the principal axis, vertices, endpoints of the minor axis, lengths of the major and minor axes, foci, eccentricity

and equations of directrices. Draw also a sketch of the ellipse.

116y

4x 22

=+

SOLUTION

From the standard equation,

.2band4a ==

Solving for the value of c,

222 bac −=

416c2 −=

3212c ==

116y

4x 22

=+

T h e r e f o r e ,

• principal axis: • vertices:

• endpoints of the minor axis

• foci

• eccentricity

• equations of the directrices given by

1164

22=+ yx

y-axis

( ) ( )4,04,0 −and( ) ( )0,20,2 −and

( ) ( )32,032,0 −and• length of the major axis• length of the minor axis

84

432=

ac

338

32162

±=±=±=cay

(2,0)(-2,0)

(0,4)

(0,-4)

( )32,0

( )32,0 −

338y =

338y −=

1164

22

=+ yx