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10-2: Conic Sections - Circles

This guide explores the fundamental concepts of circles in geometry, including the definition of a circle as the set of all coplanar points equidistant from a fixed point (the center). We cover the standard circle equation, (x-h)² + (y-k)² = r², where (h,k) is the center and r is the radius. Additionally, we demonstrate how to graph circles using given equations by completing the square and applying transformations. Key examples illustrate finding circle equations from center and radius or endpoints of a diameter, enhancing your understanding of circular geometry.

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10-2: Conic Sections - Circles

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  1. 10-2: Conic Sections - Circles circle

  2. Circle The set of all co-planar points equidistant from a fixed point (center). radius

  3. Circle Equation: (x – h)2 + (y – k)2 = r2 (x – h)2 + (y – k)2 = r (x, y) r (h, k)

  4. Circle (x – 3)2 + (y – 5)2 = 121 x – 3 = 0 r = 121 x = 3 r = 11 11 y – 5 = 0 (3, 5) y = 5 center (3, 5)

  5. Circle (x + 5)2 + (y – 2)2 = 81 x + 5 = 0 r = 81 x = -5 r = 9 9 y – 2 = 0 (-5, 2) y = 2 center (-5, 2)

  6. Graph the following circle 9x2 + 36x + 9y2 - 18y - 10 = 89 9x2 + 36x + 9y2 - 18y = 89 + 10 Remember when completing the square, the coefficient of the squared term must be 1 9x2 + 36x + 9y2 - 18y = 99 9 (x2 + 4x + 22 ) + (y2 - 2y + (-1)2 ) = 11 + 4 + 1 (x + 2)2 + (y - 1)2 = 16

  7. Circle (x + 2)2 + (y – 1)2 = 16 x + 2 = 0 r = 16 x = -2 r = 4 4 y – 1 = 0 (-2, 1) y = 1 center (-2, 1)

  8. Graph the following circle 4x2 + 24x + 4y2 + 32y + 13 = 157 4x2 + 24x + 4y2 + 32y = 157 - 13 4x2 + 24x + 4y2 + 32y = 144 4 (x2 + 6x + 32 ) + (y2 + 8y + 42 ) = 36 + 9 + 16 (x + 3)2 + (y + 4)2 = 61

  9. Circle (x + 3)2 + (y + 4)2 = 61 x + 3 = 0 r = 61 x = -3 61 y + 4 = 0 (-3, -4) y = -4 center (-3, -4)

  10. Graph the following circle 5x2 - 80x + 5y2 + 20y - 34 = 106 5x2 - 80x + 5y2 + 20y = 106 + 34 5x2 - 80x + 5y2 + 20y = 140 5 (x2 - 16x + (-8)2 ) + (y2 + 4y + 22 ) = 28 + 64 + 4 (x - 8)2 + (y + 2)2 = 96

  11. Circle (x - 8)2 + (y + 2)2 = 96 x - 8 = 0 r = 96 x = 8 r = 4 6 4 6 y + 2 = 0 (8, -2) y = -2 center (8, -2)

  12. Graph the following circle 2x2 + 10x + 2y2 + 8y + 4 = 25 2x2 + 10x + 2y2 + 8y = 25 - 4 2x2 + 10x + 2y2 + 8y = 21 2 (x2 + 5x + ) + (y2 + 4y + 22 ) = + + 4 (x + )2 + (y + 2)2 =

  13. Circle (x + )2 + (y + 2)2 = x + = 0 r = x = r = y + 2 = 0 y = -2 center

  14. Find the equation of the circle such that the endpoints of a diameter are (2,7) and (-6, 15). Center: use midpoint formula radius: use distance formula (h,k) = radius

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