Lecture Notes: Circles and the Distance Formula

Introduction

• In this lesson, we focus on the relationship between circles and the distance formula.
• A circle can be represented by a formula that relates its radius and the coordinates of its center.

Circle and Distance Formula

• Center of the circle is initially positioned at the origin (0,0).
• Radius (r) is the distance from the center to any point on the circle.
• Relation to Pythagorean theorem: For a point (x, y) on the circle, the radius can be found using:
  • [ x^2 + y^2 = r^2 ]

General Formula for a Circle

• Formula for a circle centered at (h, k) is:
  • [ (x - h)^2 + (y - k)^2 = r^2 ]
• This form is derived by moving the center from the origin to any point (h, k).

Exercises

• Identifying Center (h, k):

  • Use the formula [ x - h ] and [ y - k ] to find the center.
  • Often the center coordinates are the opposite sign of what's in the equation.

• Determining Radius (r):

  • Radius is the square root of the number on the right side of the equation.

Graphing a Circle

1. Locate the center on the graph.
2. Measure the radius from the center in north, south, east, and west directions.
3. Connect these points to form a circle.

Completing the Square

• Purpose: Useful in converting equations to the circle form.
• Steps:
  1. Group x's and y's together.
  2. Complete the square for both groups.
  3. Adjust both sides of the equation for balance.
  4. Factor the completed squares.
• Example: From general form to circle form.

Example Problems

• Convert expanded form equations into circle form by:
  • Completing the square.
  • Factoring.
  • Identifying the center and radius.

Special Case: Diameter

• Given endpoints of a diameter, find the circle's equation.
• Steps:
  1. Calculate midpoint to find the center.
  2. Use the distance formula to find the diameter, then the radius.
  3. Insert center and radius into the circle formula.

Key Concepts

• The radius can also be found directly between the center and any point on the circumference, not just by halving the diameter.
• Understanding both forms of equations helps in graphing and solving real-world problems.

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Conclusion

• The lesson ties together concepts of distance formula, midpoints, and equation transformation for circles.
• Mastery of these concepts aids in understanding geometric transformations and graphical representations of circles.