Post on 22-Mar-2018
Algebra II
Chapter 10 Conics
Notes Packet
Student Name_________________
Teacher Name_________________
1
Conic Sections
2
Identifying Conics
Ave both variables squared?'No
PARABOLA
y = a(x- h)z + k
x = a(y- k)z + h
YEs
Are the coef,flclelats of the squaredferms equal?
YEs
YES
Put l'h¢ squared !'erms together onthe same side of fhe equal sign, Areboth squ'aPed 'ÿePms positive?
HYPERBOLA
(x - fO z (y - l,)zaZ b$ = 1
(y- tOz (x-h)z l
ELLIPSE
(x-h)z (y-k)zaZ + ÿ--= I
(x-h)z (y-k)zbz ÿ a-T---= 1
rCIRCLE
(x - h) Z + (y _ k) z = rz
3
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o
X 0
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4
Circles
STANDARD FORM:
CENTER:
RADIUS:
I. Rewrite in standard form. State the center and radius. Graph the circle.
1. 𝑥2 + 𝑦2 − 4𝑥 − 16𝑦 + 64 = 0 2. 𝑥2 + 𝑦2 + 6𝑥 − 2𝑦 − 26 = 0
Center:_________ Center:_________
Radius:_________ Radius:_________
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Distance Formula: Midpoint Formula:
II. Write the equation for the following circles in standard form.
1. Center (−3, 2) and radius 3.
2. Center (2, −1) and goes thru point (5, 4).
3. Endpoints of the diameter are (10, 4) and (2, 4).
III. Given the following circles, write the equation in standard form.
1. _______________________ 2. _______________________
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Parabolas—Notes
2 forms:
1. y = a(x-h)2 + k
Opens up or down. If “a” is positive, it opens up. If “a” is negative, it opens
down.
Vertex is at (h, k)
2. x = a(y-k)2 + h
Opens left or right. If “a” is positive, it opens to the right. If “a” is
negative, it opens to the left.
Vertex is at (h, k)
1
4a
p “p” is the distance from the vertex to the focus.
The AOS and the directrix are written as EQUATIONS!!!!
The focus is located inside the parabola on the axis of symmetry.
THE VERTEX IS HALFWAY BETWEEN THE FOCUS AND THE
DIRECTRIX!!!!!!!
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Directions for graphing a parabola:
1. Find and plot the vertex.
2. Decide how the parabola opens (up/down/left/right)
3. Find the “p” value. It is found by using the following: 1
4a
p . The “p” value is the
distance from the vertex to the focus. (It is also the distance from the vertex to
the directrix.)
4. Count and plot the focus. It is a point INSIDE the parabola
5. Count and plot the directrix. It is a line outside of the parabola. It NEVER touches
the parabola.
6. Plot at least 2 points on each side of the vertex and sketch the parabola
I. GRAPHING:
1. 3)2(4
1 2 xy
Opens_______________
Vertex______________
“a”__________ “p”___________
Focus_______________
Directrix____________
AOS________________
8
2. 4)2(16
1 2 yx
Opens_______________
Vertex______________
“a”_____________ “p”____________
Focus_______________
Directrix____________
AOS________________
3. 2)3(8
1 2 yx
Opens_______________
Vertex______________
“a”_____________ “p”____________
Focus_______________
Directrix____________
AOS________________
II. Write the equation of each parabola with the given information.
1. Vertex (2, 3) and focus (0, 3)
2. Directrix: y = -5 and Focus (2, 1)
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III. Write the equation of the parabolas below:
X
Y
X
Y
IV. Rewrite each equation in vertex form. Fill in the blanks.
1. 2 10 21x y y 2. 2 2 8y x x 3. 2 4 4 16 0y x y
Vertex___________ Vertex______________ Vertex______________
Opens____________ Opens_______________ Opens_______________
“a”_______ “p”______ “a”_______ “p”_______ “a”_______ “p”_______
Focus____________ Focus____________ Focus____________
Directrix_________ Directrix_________ Directrix_________
AOS____________ AOS____________ AOS____________
10
ELLIPSES—NOTES
𝑐2 = 𝑎2 − 𝑏2 (this helps find the foci)
Ellipses always = 1
HORIZONTAL VERTICAL
Pictures:
Standard Form:
Center:
Vertices:
Co-Vertices:
Foci:
Major Axis:____________________________________________________
Minor Axis:____________________________________________________
Vertices:______________________________________________________
Co-Vertices:___________________________________________________
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Directions for graphing an ellipse:
1. Find and plot the center.
2. Decide if the major axis is vertical or horizontal. If the number under 2x is the larger
number, it is horizontal. If the number under 2y is larger, it is vertical.
3. Take the square root of the larger number ( 2a ). Count that number of spaces from the
center in the direction of the major axis.
4. Do the same for the minor axis except use the square root of the smaller number ( 2b ).
5. Sketch in the ellipse.
6. 𝑐2 = 𝑎2 − 𝑏2. This will help you find the foci(located inside the ellipse). The “c”
value is the distance from the center to the foci. Count that number of spaces from
the center and plot the foci. The foci are located on the major axis.
7. To find the coordinates of the foci, add and subtract the “c” value from the x-
value of the center if the ellipse is horizontal and from the y-value if the ellipse is
vertical.
Write in standard form (if necessary). Find the center, a, b, c values, the vertices, co-vertices,
foci, and the lengths of each axis. Then graph the ellipse.
1. 2 2( 4) ( 3)
125 9
x y
Center:_________
a=____, b=_____, c=_____
Vertices:_________________
Co-vertices:_______________
Foci:_____________________
MA=_________ ma=_________
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2. 2 2( 2) ( 2)
19 25
x y
Center:_________
a=____, b=_____, c=_____
Vertices:_________________
Co-vertices:_______________
Foci:_____________________
MA=_________ ma=_________
3. 14
)3(
36
)1( 22
yx
Center:_________
a=____, b=_____, c=_____
Vertices:_________________
Co-vertices:_______________
Foci:_____________________
MA=_________ ma=_________
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4. 25x 2 + 9y 2 = 225
Center:_________
a=____, b=_____, c=_____
Vertices:_________________
Co-vertices:_______________
Foci:_____________________
MA=_________ ma=_________
5. 25x 2 + 16y 2 - 50x – 128y -119 = 0
Center:_________
a=____, b=_____, c=_____
Vertices:_________________
Co-vertices:_______________
Foci:_____________________
MA=_________ ma=_________
Write the equation for each ellipse. 14
6. Length of major axis is 14. Foci (4,0) and (-4, 0)
7. Vertices: (2, 8) and (2, 0). Co-vertices (5, 4) and (-1, 4).
8. MA endpoints (5, 10) & (5, 0); ma endpoints (3, 7) & (-1, 7)
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Hyperbolas
𝒂𝟐 𝒊𝒔 𝑨𝑳𝑾𝑨𝒀𝑺 𝒖𝒏𝒅𝒆𝒓 𝒕𝒉𝒆 𝑷𝑶𝑺𝑰𝑻𝑰𝑽𝑬 𝒕𝒆𝒓𝒎‼!
𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐 𝑻𝒉𝒊𝒔 𝒉𝒆𝒍𝒑𝒔 𝒚𝒐𝒖 𝒇𝒊𝒏𝒅 𝒕𝒉𝒆 𝒇𝒐𝒄𝒊
16
17
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Hyperbolas Day 2
Transverse Axis:______________________________________________
Write the equation of the hyperbola that satisfies the given conditions.
Remember, you need the center, and values of 2a and 2b
1. Center (2, 2), transverse axis parallel to x-axis, a focus at (10, 2) and a
vertex at (5, 2).
2. Center at (-2, 2), a vertex at (-2, -4), a focus at (-2, -6), transverse axis
parallel to y-axis.
3. Foci at (4, 0) and (-4, 0), length of the transverse axis is 2
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Write in standard form; then find all parts:
4. 2 29 4 18 24 63 0x y x y
5. 2 216 4 96 40 108 0x y x y
20
Classifying a Conic from its General Equation
The graph of 𝐴𝑥2 + 𝐶𝑦2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 is the following:
1. Circle: 𝐴 = 𝐶
2. Parabola: 𝐴𝐶 = 0 𝐴 = 0 𝑜𝑟 𝐶 = 0, 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑏𝑜𝑡ℎ
3. Ellipse: 𝐴𝐶 > 0 𝐴 𝑎𝑛𝑑 𝐶 ℎ𝑎𝑣𝑒 𝑙𝑖𝑘𝑒 𝑠𝑖𝑔𝑛𝑠
4. Hyperbola: 𝐴𝐶 < 0 𝐴 𝑎𝑛𝑑 𝐶 ℎ𝑎𝑣𝑒 𝑢𝑛𝑙𝑖𝑘𝑒 𝑠𝑖𝑔𝑛𝑠
The test above is valid if the graph is a conic. The test does not apply
to equations such as 𝑥2 + 𝑦2 = −1, whose graph is not a conic.
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Identify the following equations as being linear, parabola, circle, ellipse, or hyperbola.
1. (𝑥+3)2
36+
(𝑦−4)2
25= 1
2. 𝑦2 + 5 = 2(𝑥 + 6)
3. 3𝑥 − 4𝑦 = −12
4. 3𝑥2 + 3𝑦2 = 48
5. 4𝑥2 − 𝑦2 + 24𝑥 + 32 = 0
6.
2 2( 2)1
36 25
x y
7.
2 2( 4) ( 2)1
5 5
x y
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Classify each conic section and write the equation in standard form.
1. −𝑦2 + 𝑥 + 8𝑦 − 17 = 0
2. 𝑥2 + 𝑦2 + 6𝑥 − 2𝑦 + 9 = 0
3. 9𝑥2 + 4𝑦2 − 54𝑥 − 8𝑦 − 59 = 0
4. −9𝑦2 + 25𝑥2 − 100𝑥 − 125 = 0
5. 4𝑥2 + 4𝑦2 − 20𝑥 − 32𝑦 + 81 = 0
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