introduction to conics conic section: circle conic section: ellipse conic section: parabola
TRANSCRIPT
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