Section 8.1: Conic Sections & Parabolas
Homework: 1, 3, 8, 11, 14, 15, 22, 25, 33, 34,
39,44,47,51,54,55,61,66,67
**Conic Sections
I
Elli . >
**Geometrv of a Parabola
Poi nl on .he p;!r.lbob
Di t.
10 drreClrlX
.>
Dir('Cln'(
Parabolas with Vertex (0, 0)
( x~
• Standard e{luation
41'0
=
C\"~ =
To the righ t or
to the left
pward or
• O l)('n,
41'V
dO\~nward
• Foeu.,
(0 . p)
(p.OI
• Di n'd rix
y =-p
.t =-p
• .\xb
"-ax is
x-axis
• Fo('al I(' ngth
I'
p
• Focal \\ id lh
l-t/' I
i.:Jpl
Gx- fII~
=
~p(y
-
ki) ~Y - kf
=
+ pI
(II
Ii:;.
p. k)
Foell '
(h. k
Di re('t r ix
y=k-p
x=h- I'
• . \xi ~
x=h
,. =
·
Foca I lengt II
p
I'
FO{,<l l \\ idlh
1-t1' 1
I~I' I
·
~p(x -
To the right or
to the left
Upward or
dowm\ard
• Opelh
·
·
/
WIth Vertex (h, kl
Parabol
• Stand ard {'(Ill ation
T
k
- - --.--tI:~---
In exercises 1-6, find the vertex, focus, directrix, and focal width of the parabola.
2. (x
+ 4)2 = -6(y + 1)
-~ ~
L\
f
-V~rtex:: (-4,-1)
- f 0 CU s." {h , \\ t p)
Ver-tex
- focus
(0 10)
~ (0, - liz)
- dI (tci." '1:: - (- 'Iz )
: '12
- focal
\A}\d1h::.
~
-: (-4,
it t-IJ) : : l-4, -Sfz.)
- direct:: Y= k-P
::
- \ 13
/1. = il2.
\4- (-Ill) I
'2
In exercises 11-30, find an equation in standard form for the parabola that satisfies the given conditions.
3. vertex (0,0), focus (-3,0)
~
l05J}
----+-;;...-;.­
f= - 3
2. vertex 3, 4),focus (5,4) l~-\c.) ;J.
= ~p
lX-h)
h1 (US = (5 I '1)
(hip, K )
j
1 :.
~ ( _~) X
t3
L1)
1 __
/
~ 1;;
\1x s. f DeuS ( - 2,0), directrix x =
6. vertex (4,3), directrix x = 6
2
vtr1e)(
IS
J-OC\)S ~
d lrectrl X
fa C IJ 5 =
(p
In
I
0
bet~f\
QlrecinK ==> ~ -- \\-p
~ ~
)~ p " - 2
4 -p
P-=-l
L
~ '"4(-2)X
~
2
(~ -3) = - 8 ( X-1 )
2
: : -8x
In exercises, 31-31: graph by hand. In exercises 37-48: use a calculator.
8. (y - 4)2 = 8(x - 3)
~-~ :: !f~(l(-3)'
~ = ~~8(x-3)t t4
~ ~ ~8(X-3)
I
ii
1
Examples: #52 & #54
@)
('1- K) 1 " 4p ('l( - h)
~1-2j ,.4x -\1:.-0
~l - l~,.-L
==
-1)(+12
(~_\)l", - 4XTi3
(~-Il ·l ~ -4(x - ~)
\jert~x. ~
(~/I)
50 (US -:: ( h T P I
( ~ -I,
~ (-~ II)
dl rect . ~ x.":: h - P
"" ~ Ti
~
t-L
the
,!:,cLva 'fe)
(~t3)1 '-
-;: ill
4p (",-I)
X -= 5 .5
"1-=0
(oT3}4":- 4iJ(5 S-I)
9": 15 P
p-= ~
k )
I)
l(,otY"lplet(:"
~ 2 = -~B(~ -3)\ i 1