Conic Sections and Their Equations

Definition And Types

• Conic Sections: Curves obtained by intersecting a plane with a double-napped cone.
• Types of Conic Sections:
  1. Circle: All points equidistant from a center point.
  2. Ellipse: Set of points where the sum of distances to two foci is constant.
  3. Parabola: Set of points equidistant from a fixed point (focus) and a fixed line (directrix).
  4. Hyperbola: Set of points where the absolute difference of distances to two foci is constant.

Equations Of Conics

• General Form: Ax² + Bxy + Cy² + Dx + Ey + F = 0
• Specific Forms:
  • Circle: (x - h)² + (y - k)² = r²; center (h, k), radius r.
  • Ellipse: (x - h)²/a² + (y - k)²/b² = 1; a,b = semi-major and semi-minor axes.
  • Parabola: y = ax² + bx + c (horizontal) or x = ay² + by + c (vertical).
  • Hyperbola: (x - h)²/a² - (y - k)²/b² = 1 (horizontal) or (y - k)²/a² - (x - h)²/b² = 1 (vertical).

Graphing Conic Sections

• Circle: Center at (h, k) with radius r; symmetrical about center.
• Ellipse: Determine vertices and co-vertices based on lengths a and b; elongated along the major axis.
• Parabola: Identify vertex, focus, and directrix; opens upwards/downwards or sideways.
• Hyperbola: Identify center, vertices, foci; asymptotes guide shape.
• Key Points for Graphing:
  • Identify conic type from the equation.
  • Calculate critical points (center, vertices, foci).
  • Draw asymptotes for hyperbolas.
  • Use symmetry as a guide for sketching curves.

Definition And Types

• Conic Sections: Curves formed by the intersection of a plane with a double-napped cone.
• Circle: Defined as the set of all points equidistant from a center point.
• Ellipse: Comprised of points where the sum of distances to two focal points remains constant.
• Parabola: Consists of points equidistant from a fixed point (focus) and a fixed line (directrix).
• Hyperbola: Characterized by points where the absolute difference in distances to two foci is constant.

Equations Of Conics

• General Form of Conics: Expressed as Ax² + Bxy + Cy² + Dx + Ey + F = 0.
• Circle Equation: (x - h)² + (y - k)² = r², where (h, k) denotes the center and r is the radius.
• Ellipse Equation: (x - h)²/a² + (y - k)²/b² = 1, defining semi-major axis a and semi-minor axis b.
• Parabola Equations:
  • Horizontal form: y = ax² + bx + c
  • Vertical form: x = ay² + by + c.
• Hyperbola Equations:
  • Horizontal: (x - h)²/a² - (y - k)²/b² = 1
  • Vertical: (y - k)²/a² - (x - h)²/b² = 1.

Graphing Conic Sections

• Circle Graphing: Place at center (h, k) and draw radius r; exhibits symmetry.
• Ellipse Graphing: Determine vertices and co-vertices based on lengths of semi-major axis (a) and semi-minor axis (b); elongated along the major axis.
• Parabola Graphing: Locate vertex, focus, and directrix; can open upwards, downwards, or sideways.
• Hyperbola Graphing: Identify center, vertices, and foci; utilize asymptotes to influence shape.
• Key Points for Graphing:
  • Recognize conic type via its equation.
  • Compute critical points like center, vertices, and foci.
  • Draw asymptotes for hyperbolas.
  • Employ symmetry principles to assist in drafting the curves.