Grade 12 Analytical Geometry - Circle Equations

Introduction to Circle Equations

• Previous topics included parabolas and hyperbolas.
• A circle is another type of graph defined by its own equation.

Circle Equation Structure

• The general equation of a circle:
  • (x - a)² + (y - b)² = r²
    • a: x-coordinate of the center
    • b: y-coordinate of the center
    • r: radius of the circle

Example 1: Circle with Specific Center and Radius

• Given:
  • Center: (2, 4)
  • Radius: 7
• Equation of the circle:
  • (x - 2)² + (y - 4)² = 7²
  • (x - 2)² + (y - 4)² = 49

Example 2: Circle in Different Quadrant

• Given:
  • Center: (-2, 4)
  • Radius: 7
• Equation of the circle:
  • (x + 2)² + (y - 4)² = 7²
  • (x + 2)² + (y - 4)² = 49

Example 3: Circle at the Origin

• Given:
  • Center: (0, 0)
  • Radius: 5
• Equation of the circle:
  • x² + y² = 5²
  • x² + y² = 25

Example 4: Circle from Diameter Coordinates

• Given diameter points A and B:
  • A = (7, 10)
  • B = (11, -2)
• Find the center using the midpoint formula:
  • Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
  • Center = ((7 + 11)/2, (10 - 2)/2) = (9, 4)
• Radius calculation:
  • Distance formula used to find length AB:
  • Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
  • Distance = √((7 - 11)² + (10 - (-2))²)
  • Calculate to find length and radius:
  • Length = 4√10, Radius = 2√10
  • Radius squared = (2√10)² = 40
• Equation of the circle:
  • (x - 9)² + (y - 4)² = 40

Example 5: Circle with Center and Radius

• Given:
  • Center: (-2, 7)
  • Radius: 8
• Equation of the circle:
  • (x + 2)² + (y - 7)² = 8²
  • (x + 2)² + (y - 7)² = 64

Summary

• Remember to use opposite coordinates for the circle center in the formula.
• The equation of a circle can be adapted based on its center and radius.