Graphing Conic Sections: Circles, Ellipses, Parabolas, and Hyperbolas

Introduction

Standard Equation:
- Horizontal: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)
- Vertical: \( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \)
- a is always greater than b
Key Points:
- Major axis: 2a
- Minor axis: 2b
- Vertices at \( (h \pm a, k) \) or \( (h, k \pm a) \)
- Foci: \( c^2 = a^2 - b^2 \), \( (h \pm c, k) \) or \( (h, k \pm c) \)
Examples:
1. \( \frac{x^2}{25} + \frac{y^2}{49} = 1 \)
  - Center: (0, 0)
  - Major axis: 14
  - Minor axis: 10
  - Foci: \( (\pm 2\sqrt{6}, 0) \)
2. \( \frac{(x-2)^2}{16} + \frac{(y+3)^2}{9} = 1 \)
  - Center: (2, -3)
  - Major axis: 8
  - Minor axis: 6
  - Foci: \( (2 \pm \sqrt{7}, -3) \)
3. \( \frac{(x+1)^2}{9} + \frac{(y-2)^2}{25} = 1 \)
  - Center: (-1, 2)
  - Major axis: 10
  - Minor axis: 6
  - Foci: \( (-1, 2 \pm \sqrt{16}) \)

Standard Equation:
- Horizontal: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \)
- Vertical: \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \)
Key Points:
- Vertices at \( (h \pm a, k) \) or \( (h, k \pm a) \)
- Foci: \( c^2 = a^2 + b^2 \), \( (h \pm c, k) \) or \( (h, k \pm c) \)
- Asymptotes: \( y = k \pm \frac{b}{a}(x-h) \) or \( y = k \pm \frac{a}{b}(x-h) \)
Examples:
1. \( \frac{x^2}{9} - \frac{y^2}{4} = 1 \)
  - Center: (0, 0)
  - Vertices: \( (\pm 3, 0) \)
  - Foci: \( (\pm \sqrt{13}, 0) \)
  - Asymptotes: \( y = \pm \frac{2}{3}x \)
2. \( \frac{(x+1)^2}{25} - \frac{(y-2)^2}{16} = 1 \)
  - Center: (-1, 2)
  - Vertices: \( (-6, 2), (4, 2) \)
  - Foci: \( (-1 \pm \sqrt{41}, 2) \)
  - Asymptotes: \( y-2 = \pm \frac{4}{5}(x+1) \)
3. \( \frac{(y-2)^2}{4} - \frac{(x+3)^2}{9} = 1 \)
  - Center: (-3, 2)
  - Vertices: \( (-3, 4), (-3, 0) \)
  - Foci: \( (-3, 2 \pm \sqrt{13}) \)
  - Asymptotes: \( y-2 = \pm \frac{2}{3}(x+3) \)

Standard Equation:
- Horizontal: \( (y-k)^2 = 4p(x-h) \)
- Vertical: \( (x-h)^2 = 4p(y-k) \)
Key Points:
- Opens right if p > 0, left if p < 0
- Opens up if p > 0, down if p < 0
- Focus: (h+p, k) or (h, k+p)
- Directrix: x=h-p or y=k-p
Examples:
1. \( y^2 = 8x \)
  - Vertex: (0, 0)
  - Focus: (2, 0)
  - Directrix: x=-2
2. \( (y-2)^2 = 4(x-3) \)
  - Vertex: (3, 2)
  - Focus: (4, 2)
  - Directrix: x=2
3. \( (x+1)^2 = -2(y-3) \)
  - Vertex: (-1, 3)
  - Focus: (-1, 5.5)
  - Directrix: y=2.5

Circle: \( A = C \)
- Example: \( x^2 + 4y^2 + 8x = 0 \)
Ellipse: \( A \neq C \), both positive
- Example: \( 4x^2 + 25y^2 = 100 \)
Hyperbola: \( A \) positive, \( C \) negative (or vice versa)
- Example: \( 4x^2 - 9y^2 = 36 \)
Parabola: Only one squared term
- Example: \( x^2 - 8x = 4y \)

Circles:
- Example: \( 2x^2 + 8x + 2y^2 + 4y = 6 \)
  1. Group like terms: \( 2(x^2 + 4x) + 2(y^2 + 2y) = 6 \)
  2. Complete the square: \( 2((x+2)^2 - 4) + 2((y+1)^2 - 1) = 6 \)
  3. Simplify: \( (x+2)^2 + (y+1)^2 = 8 \)
Ellipses:
- Example: \( 4x^2 + 25y^2 - 24x + 100y = -36 \)
  1. Group like terms: \( 4(x^2 - 6x) + 25(y^2 + 4y) = 36 \)
  2. Complete the square: \( 4((x-3)^2 - 9) + 25((y+2)^2 - 4) = 36 \)
  3. Simplify: \( \frac{(x-3)^2}{25} + \frac{(y+2)^2}{9} = 1 \)
Hyperbolas:
- Example: \( 4x^2 - 9y^2 - 16x + 54y = 101 \)
  1. Group like terms: \( 4(x^2 - 4x) - 9(y^2 - 6y) = 36 \)
  2. Complete the square: \( 4((x-2)^2 - 4) - 9((y-3)^2 - 9) = 36 \)
  3. Simplify: \( \frac{(x-2)^2}{9} - \frac{(y-3)^2}{4} = 1 \)
Parabolas:
- Example: \( x^2 + 6x = 4y - 1 \)
  1. Complete the square: \( (x+3)^2 - 9 = 4y - 1 \)
  2. Simplify: \( (x+3)^2 = 4(y-1/4) \)