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

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

2. 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) Major axis Since a2 = 25, then a =  5 Since b2 = 16, then b =  4 Minor axis Major: (7 + 5, -2)  (12, -2) (7  5, -2)  (2, -2) Minor: (7, -2 + 4)  (7, 2) (7, -2  4)  (7, -6)

5. Example: Graph the following equation: 4x2 + 25y2 = 400 Put in standard form 400 400 400 x2 + y2 = 1 100 16 Center: (0, 0) Major axis Since a2 = 100, then a =  10 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 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. Putting Equations in Standard Form by Completing the Square Example: Graph the following equation: 4y2 + 9x2 – 24y – 72x + 144 = 0 Group the x terms and y terms 9x2 – 72x + 4y2 – 24y = - 144 Factor and leave blanks to complete the square: 9(x2 – 8x + ) + 4(y2 – 6y + ) = - 144 + + 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 + 144 = 0 Put in standard form 9(x – 4)2 + 4(y – 3)2 = 36 36 36 36 (x – 4)2 + (y – 3)2 = 1 4 9 Center: (4, 3) Minor axis Since a2 = 4, then a =  2 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 + ) = - 44 + + ) 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, -3 + 2)  (2, -1) (2, -3 – 2)  (2, -5)