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Circle

Conic Sections. Circle. Ellipse. Hyperbola. Parabola. Ex. 1. Circles. x 2 + y 2 = 16. 4. 4. 4. 4. Standard form: (x – h) 2 + (y – k) 2 = r 2. (x – 0) 2 + (y – 0) 2 = 4 2. center: (h, k). radius: r. (0,0). center:. radius:. 4. 6. x 2 + y 2 = 36. (0,0). center:.

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Circle

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  1. Conic Sections Circle Ellipse Hyperbola Parabola

  2. Ex. 1 Circles x2 + y2 = 16 4 4 4 4 Standard form: (x – h)2 + (y – k)2 = r2 (x – 0)2 + (y – 0)2 = 42 center: (h, k) radius: r (0,0) center: radius: 4

  3. 6 x2 + y2 = 36 (0,0) center: radius: 6

  4. 2 15 60 7.746 x2 + y2 = 60 (0,0) center: radius:

  5. 3 (x-5)2 + (y+2)2 = 9 (5,-2) center: radius: 3

  6. Find the center and radius of the circle: x2 + y2 + 10x – 6y + 33 = 0 + 9 x2 + 10x + y2 – 6y = -33 + 25 + 25 + 9 Fill in the blanks by completing the square If you add 25 and 9 to one side, you must add 25 and 9 to the other side Factor the left hand side (x + 5)2 + (y – 3)2 = 1 Center: (-5, 3) 1 Radius:

  7. Find the center and radius of the circle: x2 + y2 + 8x – 2y + 13 = 0 + 1 x2 + 8x + y2 – 2y = -13 + 16 + 16 + 1 Fill in the blanks by completing the square If you add 16 and 1 to one side, you must add 16 and 1 to the other side Factor the left hand side (x + 4)2 + (y – 1)2 = 4 Center: (-4, 1) 2 Radius:

  8. x2 y2 25 4 Ellipse + = 1 (x - h)2 (y - k)2 a2 b2 Standard form: + = 1 2 5 5 2 Center = (0,0) Major Axis = 10 Minor Axis = 4

  9. x2 y2 25 4 Ellipse + = 1 (x - h)2 (y - k)2 a2 b2 Standard form: + = 1 2 5 5 21 -21 2 21 = c Foci: (021, 0) Center = (0,0) Foci will lie on the major axis c units from the center Major Axis = 10 Minor Axis = 4 a2 – b2 = c2 25 – 4 = c2 21 = c2

  10. 36x2 +9y2 – 72x +54y = 207 6 (x – 1)2 (y + 3)2 9 36 3 3 + = 1 6 36x2 – 72x + 9y2 +54y = 207 +36 + 81 +1 +9 36(x2 – 2x ) + 9(y2 + 6y ) = 207 36(x – 1)2 + 9(y + 3)2 =324 324 324 Center = (1, -3) Major Axis = 12 Minor Axis = 6

  11. >The distance to the foci from the center is c. 27 = c >The foci are at the point (1, -3  27) 27 27 Finding the Focus (foci) (x-1)2 (y+3)2 9 36 + = 1 >The foci lie on the major axis a2 – b2 = c2 36 – 9 = c2 27 = c2

  12. x2 y2 25 1 Center = (0,0) + = 1 (0  26, 0) 24 or 2 6 = c Major Axis = 10 Minor Axis = 2 Foci: a2 – b2 = c2 Foci: 25 – 1 = c2 24 = c2

  13. (x-2)2 (y+3)2 9 25 + = 1 10 6 5 3 3 5 Center = (2,-3) Major Axis = Minor Axis =

  14. (x-2)2 (y+3)2 9 25 + = 1 10 6 5 3 3 5 Center = (2,-3) Major Axis = Minor Axis = Foci: 25 – 9 = c2 16 = c2 4 = c Foci are at (2, -3  4)

  15. (x-1)2 (y+3)2 9 36 (x-1)2 (y+3)2 9 36 (x-1)2 (y+3)2 936 + = 1 + = 1 + = 1 6 3 3 6 Moving the center >The center will be at (1, -3) >The major axis will be 12 Units long and be parallel to The y-axis. >The minor axis will be 6 units long and be parallel to the x-axis.

  16. (x + 4)2 (y - 1)2 36 7 + = 1 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 = 252 252 252

  17. Center = (-4, 1) (x+4)2 (y-1)2 36 7 + = 1 2 7 -29 +29 6 6 7 7 (-4 29, 1) 29 = c Major Axis = 12 Minor Axis = Foci: a2 - b2 = c2 Add c to the x of the center 36 - 7 = c2 29 = c2

  18. x2 y2 9 36 Hyperbola - = 1 Center: (0,0) Out on the x-axis 3 and -3 Out on the y-axis 6 and -6 Draw asymptotes through corners of box slope = ± 6/3 or ± 2/1 Draw hyperbola on the x-axis not crossing the dotted lines.

  19. x2 y2 9 36 - = 1 45 = c (0 45, 0) Foci: a2 + b2 = c2 9 + 36 = c2 45 = c2

  20. (y + 3)2 (x - 1)2 25 9 - = 1 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 = 225 225 225

  21. (y+3)2 (x-1)2 25 9 - = 1  34 = c Foci: (1,-3 34) Center : (1, -3) Slopes of Asymptotes : 5/3 Foci : a2 + b2 = c2 25 + 9 = c2 34 = c2

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