Introduction to Conics

Conic Section: Circle

Conic Section: Ellipse

Conic Section: Parabola

Conics

r(x,y)

(h,k)

What are Conic Sections? 2 videos

1) Introduction to Conics (8 min.)Videos\Intro to Conics+Circles

\Introduction to Conic Sections A.rv

2) Introduction to Circles (8:15 min.)Videos\Intro to Conics+Circles

\Conic Sections Intro to Circles B.rv

Circle is All points equidistant, r, from a single point, the center.

• Standard “center radius” form of a Circle?

Center at (0,0)

Center at (h,k)

What are Conic Sections?What is the standard form for a Circle?

r

(0,0)

(x,y)

Examples follow 1)()(

2

2

2

2

r

ky

r

hx

222 )()( rkykx

Examples:1)Center (0,0) and radius 3 (x-0)2+(y-0)2=9

x2+y2=9

2)Center (0,2) and radius 11 (x-0)2+(y-2)2 = 121

3)Center (3,4) and radius 2 (x-3)2+(y-4)2 = 4

4)Center (-4,0) and radius 5 (x+4)2+(y-0)2 = 25

5) Center (0,0) and radius 1/2 (x-0)2+(y-0)2 = 1/4

32

Write the standard equation:

a)Center (0,0) and radius 6

b)Center (0,0) and radius 9

c)Center (0,0) and radius 11

d)Center (0,0) and radius 5

e)Center (2,0) and radius 6

f) Center (3,0) and radius 9

g)Center (0,–2) and radius

3

h)Center ( 2, 3) and radius

6

i) Center (–3, –5) and radius

5

j) Center (–11, –12) and

radius 4

Circles

What are Conic Sections? Video -- Introduction to Ellipse (13 min.)Videos\Intro to Ellipses C1.rv

Definition: All points in a plane, the sum of whose distances from two fixed points (foci) is constant.

The standard eq. form of an Ellipse

Center at (h, k)

What are Conic Sections?What is the standard form for an Ellipse?

(h,k)

1)()(

2

2

2

2

b

ky

a

hx

( h+a, k)

( h–a, k)

( h, k–b)

( h, k+b)

Sketching Ellipses:

( 4, 0)( –4, 0)

( 0, +3)

( 0, –3)

(0,0)

( 4, 0)

( –4, 0)

( 0, +5)

( 0, –5)

(0,0)

( 8, 0)( –2, 0)

( 3, 6)

( 3, –2)

(3,2)

a

b

c

1916

22

yx

12516

22

yx

116

)2(

25

)3( 22

yx

Sketch the Ellipse labeling its Center & Vertices

( h+a, k)

( h–a, k)

( h, k+b)

( h, k–b)

(h,k)

(h,k)

( h+b, k)

( h, k+a)

( h–b, k)

( h, k–a)

a

b

c

d

12536

22

yx

19

22

yx

125

)2(

16

)1( 22

yx

14

)3(

9

)2( 22

yx

Homework - Sketch the Ellipse labeling its Center & Vertices

11625

22

yx

1169144

22

yx

Page 364, #s 35, 36, 39, 40, 41. 42

159

)4( 22

yx

3694 2 yx

164

)5(

28

)4( 22

yx

1

2

3

4

5

An Ellipse has 2 foci Definition (reworded): an Ellipse is the set of

points where the sum of the points’ distances from the 2 foci is a constant.

Determining the location of the 2 foci… ..\7th 5 weeks\Foci of an Ellipse C2.rv

Important relationships:

Let the focus length be equal to cc2=a2-b2

d1+d2=2aEccentricity (flatness), e = c/a,

d1 d2

a c

Ellipse -- foci

Examples follow

What is the ellipse’s equation (in standard form) given…

Vertices: (±5,7) Foci: (±3,7)c2=a2-b2

Since, a=5 & c=3, then b=4

Ellipse

a c

(3,7)(-3,7)(-5,7) (5,7)

1)()(2

2

2

2

b

y

a

x1

16

)(

25

)( 22

yx

Note: The Foci are always on the major axis !!

Vertices: (±13,1) Foci: (±12,1)

Vertices: (±4,7) Foci: (±3,7)

Vertices : (2,1), (+14,1) Foci: (4,1), (+12,1)

Vertices: (7,±5) Foci: (7,±3)

Write the Ellipse’s equation…and then ID its Eccentricity

a c

b

Sketching the ellipse first, might HELP !

Page 364, #s 47 through 50 and 51 for extra credit

Homework

“Back from Spring Break” Review

CONICS

Circles Ellipses

short & wide

tall & thin

Parabolas

up & down

right & left

(next)

How to quickly Identify the conic from the equation (future) ?

√

√ √ √

√

Circle:

Ellipse:c2=a2-b2

d1+d2=2aEccentricity (flatness), e = c/a

Parabola: if vertex is at (0,0)

if vertex is at ( h, k)

“Back from Spring Break” Review

r

(h,k)

(x,y)

1)()(

2

2

2

2

r

ky

r

hx

( h+a, k)

( h–a, k)

( h, k–b)

( h, k+b)

d1

d2

a c

1)()(

2

2

2

2

b

ky

a

hx

khxay 2)(

2)()( hxaky

2)(xay We have studied parabolas that point up or down (so far).

Circle –

“Back from Spring Break” Review

Ellipse –

Parabola –

set of all points that are the same distance (equidistant), r, from a single point, the center.

set of all points in a plane, the sum of whose distances from two fixed points (foci) is constant.

set of points in a plane that are equidistant from a fixed line (the directrix) and a point (the focus).

Define In Your W

ords

(mathematic

ally)

Parabola – opening up or down, the equation is:Point 1: And if the vertex (h,k) is at (0,0), then

becomes

Similarly if we have a parabola opening left or right then the x and y is switched around Point 2: p is the distance from the vertex

to the focus andto the directrix

Note By the definition of a parabola the vertex is always midway between the focus and the directrix.

Point 3: Hence, to find that distance divide the coefficient of the variable (the variable having a 1 as its exponent) by 4.

Finding Parabola focus when vertex is at (0,0)

khxay 2)( pyx 42

pxy 42

Reference Drawn examples on board

SUMMARIZING…

Remember the vertex is at ( 0, 0 )…if the parabola opens ‘up’ then the focus is at ( 0, p)if the parabola opens ‘down’ then the focus is at ( 0, -

p)if the parabola opens to the ‘right’ then the focus is

at ( p, 0)if the parabola opens to the ‘left’ then the focus is at

( -p, 0)

Finding Parabola focus when vertex is at (0,0)

set 4p=16 and solve for ‘p’solved… p=4therefore the focus is at ( 0, 4)

set 4p= –1/2 and solve for ‘p’solved… p= –1/8therefore the focus is at ( 0, –1/8)

set 4p=9 and solve for ‘p’solved… p = 9/4 = 2 ¼ therefore the focus is at ( 2 ¼, 0)

Find the Parabola’s focus… ypyAx )4()(2

yx2

12

yx 162

xy 92

Opens down

Opens up

Opens right

Page # 363, problem #s 1, 2, 3, 4, 11,12

Page #363, problem #s 13, 14, 15, 16

Page #363, problem #s 17, 18, 19, 20, 21, 22, 23

Homework

Due Wednesday

Due TBD

Due TBD

set 4p=16 and solve for ‘p’the focus is at ( 0, 4) -- see

previous slideand the directrix, y = –4

set 4p= –1/2 and solve for ‘p’the focus is at ( 0, –1/8)and the directrix, y = +1/8

set 4p=9 and solve for ‘p’the focus is at ( 2 ¼, 0)and the directrix, x = –2 ¼

Find the Parabola’s directrix…

yx2

12

yx 162

xy 92

Opens down

Opens up

Opens right

Eccentricity = e = c/a

Explain what the effect is on the ellipse’s shape as the focus’s distance from the center (‘c’) approaches the vertex’s distance from the center (‘a’) -- in other words, when ‘e’ approaches a value of 1.

Extra Credit – week of April 11th

DUE AT END OF PERIOD.

Background/Archive

Please note that the next school-wide writing prompt will take place on Tuesday, 4/5/11 during 2nd period.  

The prompt is as follows:"The use of Cornell Notes, Flash Cards and

Concept Maps are currently used to help you organize your notes and make your test preparation easier.  What other learning activities would you like to see incorporated in your class?"

After the essays have been completed, please compile or ask a student to make a list of the ideas submitted by your class.  Give this list to your ILT representative by the end of the day on 4/5/11. This will help the entire school!

2nd Period