Lecture Notes: Circle and Distance Formula

Introduction

• Topic: Relationship between circles and the distance formula.
• Goal: Understand how circles relate to coordinate systems and distance calculations.

Circles and Coordinate Systems

• Center of Circle: Origin (0,0) in a coordinate plane.
• Visualizing a Circle: Select a random point on the circle, draw radius from center to edge.

Radius and Distance

• Distance Formula: Originally denoted as 'd', now revised as 'r' for radius.
• Using Pythagorean Theorem:
  • Origin point (0,0), some point on circle (x1, y1).
  • Distance formula: ( \sqrt{(x2-x1)^2 + (y2-y1)^2} ).
  • For circle centered at the origin: ( x^2 + y^2 = r^2 ).

Circle Equation - Center at any Point (h, k)

• Changing Center:
  • Previous center = origin (0,0).
  • New center = (h, k).
• Circle Formula: ( (x-h)^2 + (y-k)^2 = r^2 ).

Exercises

• Identify Center and Radius:
  • Center (h, k) from equation: ( x-h ) and ( y-k ).
  • Radius r = ( \sqrt{constant} ) from the equation.
• Examples:
  • Formulas: ( x-1 ), ( y-(-4) ), etc.
  • Centers: (1, -4), (-2, 7), (-5, 0), (-3, 2).
  • Radii: 5, 3, 4, ( \sqrt{7} ).

Graphing a Circle

• Steps:
  • Identify center and radius.
  • Plot center, measure radius in all directions (N, S, E, W).
  • Draw circle through these points.
• Example: Center (-3, 2), radius ( \sqrt{7} ) (approx. 2.6).

Expanding Circle Formula

• FOIL (First, Outside, Inside, Last):
  • Convert standard circle equation by expanding.
  • Example: Expand and rearrange to standard form.
• Completing the Square:
  • Group x's and y's, move constant to other side.
  • Add square of half the coefficient to both sides.
  • Factor and simplify equation.
• Example:
  • Equation: ( x^2 + y^2 + 12x - 6y - 4 = 0 ).
  • Center: (-6, 3), Radius: 7.

Finding Circle from Diameter

• Concepts:
  • Diameter endpoints given: (-1, 3) and (3, 11).
  • Midpoint = center of circle.
• Steps:
  • Calculate midpoint for center.
  • Calculate distance for diameter, divide by 2 for radius.
  • Use distance formula for diameter or radius.
• Equation:
  • Center (h, k): Midpoint of diameter.
  • Radius: Half of diameter or distance from center to point.

Conclusion

• Connection: Ties between distance formula, midpoint, and circle equations.
• Equation Practice: Understanding circle equations through examples enhances grasp of geometry concepts.

End of Lecture: Excellent understanding of circle equations and graphical representations in coordinate systems.