1. Short Subject: Conics - circles, ellipses, parabolas, and hyperbolas

2. Example: CIRCLES General Equation: (x – h)2 + (y – k)2 = r2
where (h, k) is the center and r is the radius
Example:
Graph the following equation:
(x + 2)2 + y2 = 25
Note: Addition and Same Coefficients
center (-2,0)

3. ELLIPSES General Equation: (x – h)2 + (y – k)2 = 1 a2 b2
where (h, k) is the center.
The vertices are a units right and left from (h, k) and b units up and down from (h, k)
Center: (h, k) b
Note: Addition and Different Coefficients a a b
Major axis: 2b
since 0 < a < b (ellipse is vertical)
Minor axis: 2a

4. Example: Graph the following equation: (x – 7)2 + (y + 2)2 = 1 25 16
Center: (7, -2)
Since a2 = 25, then a =  5
Major axis
Since b2 = 16, then b =  4
Minor axis
Major:
(7 + 5, -2)  (12, -2)
(7  5, -2)  (2, -2)
Minor:
(7, )  (7, 2)
(7, -2  4)  (7, -6)

5. Example: Graph the following equation: 4x2 + 25y2 = 400 400 400 400
Put in standard form
400
400
400
x y2 = 1
100 16
Center: (0, 0)
Since a2 = 100, then a =  10
Major axis
Since b2 = 16, then b =  4
Minor axis
Major:
(0 + 10, 0)  (10, 0)
(0  10, 0)  (-10, 0)
Minor:
(0, 0 + 4)  (0, 4)
(0, 0  4)  (0, -4)

6. HYPERBOLAS I. General Equation: (x – h)2 – (y – k)2 = 1 a2 b2
where (h, k) is the center.
The vertices are (h + a, k) and (h – a, k) and opens left and right (transverse axis is horizontal).
Note: Subtraction and (+) x-coefficient
(h – a, k)
(h + a, k)
m of asymptotes =  b/a

7. HYPERBOLAS II. General Equation: (y – k)2 – (x – h)2 = 1 b2 a2
where (h, k) is the center.
The vertices are (h, k + b) and (h, k – b) and opens up and down (transverse axis is vertical).
Note: Subtraction and (+) y -coefficient
m of asymptotes =  b/a

8. Example: Graph the following equation: 25x2 – 4y2 = 400 400 400 400
Put in standard form
400
400
400
x2 – y2 = 1 16
100
m =  10/4 =  5/2
Vertices (since it opens left and right)
Center: (0, 0)
Since a2 = 16, then a =  4
Since b2 = 100, then b =  10
Vertices:
(0 + 4, 0)  (4, 0)
(0  4, 0)  (-4, 0)

9. Example: Putting Equations in Standard Form by Completing the Square
Graph the following equation:
4y2 + 9x2 – 24y – 72x = 0
Group the x terms and y terms
9x2 – 72x + 4y2 – 24y =
Factor and leave blanks to complete the square:
9(x2 – 8x ) + 4(y2 – 6y ) = 16 9
144 36
Complete the square and fill in blanks
9(x – 4)2 + 4(y – 3)2 = 36
Ellipse

10. Example (cont): Graph the following equation:
4y2 + 9x2 – 24y – 72x = 0
Put in standard form
9(x – 4)2 + 4(y – 3)2 = 36 36 36 36
(x – 4) (y – 3)2 = 1 4 9
Center: (4, 3)
Since a2 = 4, then a =  2
Minor axis
Since b2 = 9, then b =  3
Major axis
Minor:
(4 + 2, 3)  (6, 3)
(4  2, 3)  (2, 3)
Major:
(4, 3 + 3)  (4, 6)
(4, 3  3)  (4, 0)

11. Example: Graph the following equation: 12y2 – 4x2 + 72y + 16x + 44 = 0
12y2 + 72y – 4x2 + 16x = - 44
12(y2 + 6y ) – 4(x2 – 4x ) = ) 9 4
108
- 16
12(y + 3)2 – 4(x – 2)2 = 48
Put in standard form
(y + 3)2 – (x – 2)2 = 1
Hyperbola – opens up & down 4 12
Center: (2, -3)
Vertices (since it opens up & down)
Since b2 = 4, then b =  2
Since a2 = 12, then a =    3.5
Vertices:
(2, )  (2, -1)
(2, -3 – 2)  (2, -5)

12. (y + 3)2 – (x – 2)2 = 1 4 12 Put in standard form Center: (2, -3)
Vertices (since it opens up & down)
Since b2 = 4, then b =  2
Since a2 = 12, then a =    3.5
Vertices:
(2, )  (2, -1)
(2, -3 – 2)  (2, -5)
(2, -1)
(2, -3)
(2, -5)
m =  b/a