May 3, 2018

Conic Sections

10.2

- curves formed by the intersectionof a plane and a double cone

double napped come

→ -

÷÷÷¥x.. ⇒µhn÷•

← -

t,

-- - -

⇒£- t

/ t

May 3, 2018

May 3, 2018

May 3, 2018

May 3, 2018

May 3, 2018

3 asic Conic Sections-

ParabolaCircle

Ellipsehyperbola

May 3, 2018

10.2 Parabolas

Definition:

Parabola

focus

directrix

The set of all points equidistantfrom a fixed line and a

fixed point

¥ - the fixed point±.

-

'

i,

of a parabola* '

n

- the fixed lineof a parabola

May 3, 2018

Find the equation of the parabola with a focus of F(0,3) and a directerix of y = -3

→

ct-VIEEy.FI

zgEP=PQo

±÷JV¥iysI=HytI '

x' t ( y -35=0 't Gt3 )'

x' tyl-E.pt#7Eyyt9/

May 3, 2018

Find the equation for the set of all points equidistant from the point (2,0) and the line x=-2

D= Vc¥tyz

op

2 → IgFaust

directrix-

-

✓htI÷q÷E.AB -

- Bc

1*25*179" - 2) H -

ofL xt2T=fx

-277,

' I

,

"-e- ⑨

xft4xtk-H-4.ge#4ytry2←

-- no

yay rain

8x*= 4842Hey yay, hope you smiled dat !

May 3, 2018

Relating the location of the focus and directrix to the corresponding quadratic relation.

X =- zy2

X=ay2Y=a×zare parabolas

The a- value allows us to determine

the focus and directrix

Let c -

- distance from focus to vertex1ac←

T

2 → ¥

3 → I -

12

a ÷ .'

3=1 = ÷4

'

zI

May 3, 2018

Consider

If If

late F- focus D = directrix

opens up . opens down

E- ( Qc ) F-- ( O,

- C )D :y= - C D :y=cI

,Flo ,4? ye

-. I I

! i'

-

⇐- 1*-1

← y= - c f •'

,

ICope,

d '*

May 3, 2018

Consider

If If

0

opensright opens left

f- = ( GO ) F -

- C- c. O )D :X -_ - c D :X -

- c•§,

,.jFFfso,

May 3, 2018

Find the equation of a parabola with a focus of (-5,0) and a vertex at the origin.

ifa⇐ dist

.

opens-le-ft-FII.ee?nauetexII " " " "-

⇐ lxr -

Xelc •2=10--51¥,

x

a-

*

E' Inuits ,

9

May 3, 2018

Find the equation of a parabola with a

focus of and a vertex at the origin.- opens right

~IET

'

•6

plug in c to find a

f ↳ a. ÷ , . ::c:b:::S::::up to the right

Y 2

X =2J

May 3, 2018

In some solar collectors, a mirror with a parabolic cross section is used to concentrate light on a pip which is located at the focus of the mirror.

May 3, 2018

a. Suppose that the pip is located 6ft. from the vertex of the mirror. Write an equation that models the cross section of the mirror.

b. Suppose that you stretch the equation of the mirror (and the mirror itself) by a factor of 2 to become . How should you move the pipe so that it remains the focus?

c. What if the pipe is 8.25ft. from the vertex of the parabola. Write an equation to model the cross section of the mirror.

May 3, 2018

Identifying the focus and directrix from an equation.

opensdownv:( 0,0 )

I ⇐- - -- - -

- s

F :@, -4 ) %)D :y=4 T

a= -1

latte"

÷ " 9

May 3, 2018

Identifying the focus and directrix from an equation.

May 3, 2018

Identifying the vertex, focus, and directrix

AND

graphing parabolas

from equations in standard form.

⑦

f-

p-

i

-

2c{&÷÷eHorizontal

-

.£y=E,

"V

opens⑧flown

*,

*

y=ax2 Ilate V:(10,2¥)

⇐ Ef -2-4=4 V:C 10,7)⇐ fifty a= I

y = ax'

a = I

V:C10,71

Iy = a ( X - 105+7

y = ( x - lol 't 7

Vertex form of a parabola :

y = a ( x - htt K

vertex at ( h,

K )

Focus :C3-4%1 ) Directrix giftopens downvertex -17

,. to )

ly.

-

ax'

.

¥¥±w

÷ti⇒i⇒- .. .

? yeaU lat =L

✓ y -3

'

Tlate - kg

g-

acx- Afca as - Yg

yay g=-Y3Cx-as

May 3, 2018

Find the vertex, focus, and directrix.

May 3, 2018

Find the vertex, focus, and directrix.