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Circle Ellipse
Hyperbola Parabola
Conic Sections
Circles x2 + y2 = 16
center: (0,0)
radius: 4
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
44
4
4
Ex. 1
Standard form: (x – h)2 + (y – k)2 = r2
center: (h, k) radius: r
(x – 0)2 + (y – 0)2 = 42
x2 + y2 = 36
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8
center: (0,0)
radius: 6
6
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8
x2 + y2 = 60center: (0,0)
radius:
7.746
2 15
60
-6
-4
-2
0
2
-1 1 3 5 7 9
(x-5)2 + (y+2)2 = 9
center: (5,-2)
radius: 3
-6
-4
-2
0
2
-1 1 3 5 7 93
Find the center and radius of the circle: x2 + y2 + 10x – 6y + 33 = 0
x2 + 10x + y2 – 6y = -33Fill in the blanks by completing the square
+ 25 + 9
If you add 25 and 9 to one side, you must add 25 and 9 to the other side
+ 25 + 9
Factor the left hand side
(x + 5)2 + (y – 3)2 = 1
Center: (-5, 3)
Radius: 1
Find the center and radius of the circle: x2 + y2 + 8x – 2y + 13 = 0
x2 + 8x + y2 – 2y = -13Fill in the blanks by completing the square
+ 16 + 1
If you add 16 and 1 to one side, you must add 16 and 1 to the other side
+ 16 + 1
Factor the left hand side
(x + 4)2 + (y – 1)2 = 4
Center: (-4, 1)
Radius: 2
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-8 -6 -4 -2 0 2 4 6 8
Ellipse
Center = (0,0)Major Axis = 10Minor Axis = 4
x2 y2
25 4 + = 1
5 5
2
2
Standard form: + = 1(x - h)2 (y - k)2
a2 b2
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-8 -6 -4 -2 0 2 4 6 8
Ellipse
Center = (0,0)Major Axis = 10Minor Axis = 4
x2 y2
25 4 + = 1
5 5
2
2
Standard form: + = 1(x - h)2 (y - k)2
a2 b2
Foci will lie on the major axisc units from the center
a2 – b2 = c2
25 – 4 = c2
21 = c2
21 = c
Foci: (021, 0)
-21 21
36x2 +9y2 – 72x +54y = 207
Center =(1, -3)Major Axis =12Minor Axis = 6
-10
-8
-6
-4
-2
0
2
4
-4 -2 0 2 4 66
6
33
36(x2 – 2x ) + 9(y2 + 6y ) = 207
36x2 – 72x + 9y2 +54y = 207+1 +9 +36 + 81
36(x – 1)2 + 9(y + 3)2 =324324 324
(x – 1)2 (y + 3)2
9 36+ = 1
>The foci are at the point (1, -3 27)
Finding the Focus (foci)
>The foci lie on the major axis
a2 – b2 = c2
36 – 9 = c2
+ = 1(x-1)2 (y+3)2
9 36
27 = c2
27 = c
-10
-8
-6
-4
-2
0
2
4
-4 -2 0 2 4 6
27
27
>The distance to the foci from the center is c.
x2 y2
25 1 + = 1
-1
0
1
-5 -4 -3 -2 -1 0 1 2 3 4 5
Center = (0,0)Major Axis =10Minor Axis = 2
Foci: a2 – b2 = c2
25 – 1 = c2
24 = c2
24 or 2 6 = c
Foci:
(0 26, 0)
(x-2)2 (y+3)2
9 25 + = 1
Center = (2,-3)Major Axis =Minor Axis =
-10
-8
-6
-4
-2
0
2
4
-2 -1 0 1 2 3 4 5 65
5
33
-10
-8
-6
-4
-2
0
2
4
-2 -1 0 1 2 3 4 5 6
10
6
(x-2)2 (y+3)2
9 25 + = 1
Center = (2,-3)Major Axis =Minor Axis =
-10
-8
-6
-4
-2
0
2
4
-2 -1 0 1 2 3 4 5 65
5
33
-10
-8
-6
-4
-2
0
2
4
-2 -1 0 1 2 3 4 5 6
10
6
Foci: 25 – 9 = c2
16 = c2
4 = c
Foci are at (2, -3 4)
Moving the center
>The center will be at (1, -3)
-10
-8
-6
-4
-2
0
2
4
-4 -2 0 2 4 6
>The major axis will be 12Units long and be parallel toThe y-axis.
>The minor axis will be 6units long and be parallel tothe x-axis.
6
6
+ = 1(x-1)2 (y+3)2
9 36
33
+ = 1(x-1)2 (y+3)2
9 36 + = 1(x-1)2 (y+3)2
9 36
7x2 + 56x + 36y2 - 72y = 104
7x2 + 56x + 36y2 - 72y = 104
7(x2 + 8x ) + 36(y2 - 2y ) =104+ 16 + 1 +112+36
7(x + 4)2 + 36(y - 1)2 = 252252252
(x + 4)2 (y - 1)2
36 7+ = 1
(x+4)2 (y-1)2
36 7 + = 1Center =(-4, 1)Major Axis =12
Minor Axis = 2 7
-3
2
-12 -10 -8 -6 -4 -2 0 2 4
7
7
66
-3
-1
1
3
5
-12 -10 -8 -6 -4 -2 0 2 4
Foci: a2 - b2 = c2
36 - 7 = c2
29 = c2
29 = c
Add c to the x of the center
(-4 29, 1)
-29 +29
-10
-8
-6
-4
-2
0
2
4
6
8
10
-6 -4 -2 0 2 4 6
Hyperbola x2 y2
9 36 - = 1
-10
-8
-6
-4
-2
0
2
4
6
8
10
-6 -4 -2 0 2 4 6
-10
-8
-6
-4
-2
0
2
4
6
8
10
-6 -4 -2 0 2 4 6
Center: (0,0)Out on the x-axis
Out on the y-axis
3 and -3
6 and -6
Draw asymptotes through corners of
boxslope = ± 6/3 or ± 2/1
Draw hyperbola on the x-axis not
crossing the dotted lines.
-10
-8
-6
-4
-2
0
2
4
6
8
10
-6 -4 -2 0 2 4 6
Foci: a2 + b2 = c2
x2 y2
9 36 - = 1
9 + 36 = c2
45 = c2
45 = c
(0 45, 0)
9y2 + 54y - 25x2 + 50 = 169
9y2 + 54y - 25x2 + 50 = 169
9(y2 + 6y ) - 25(x2 - 2x ) =169+ 9 + 1 +81-25
9(y + 3)2 - 25(x - 1)2 = 225225225
(y + 3)2 (x - 1)2
25 9- = 1
-15
-10
-5
0
5
10
-4 -2 0 2 4 6
-15
-10
-5
0
5
10
-4 -2 0 2 4 6
(y+3)2 (x-1)2
25 9 - = 1
Center : (1, -3)Slopes of Asymptotes : 5/3
Foci : a2 + b2 = c2
25 + 9 = c2
34 = c2
34 = c
Foci: (1,-3 34)
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