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

EEI

0J

I

o

X 0

II

,b

W

tÿ

o×

0

II

+

€,l c4

H,

o

, >,, o

H

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____________

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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=_________

12

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

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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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