Lecture Notes: The Circle in Pre-Calculus

Definition of a Circle

• A circle is defined as the set of all points in a plane that are at a constant distance (radius) from a fixed point (center).
• Center: The fixed point of the circle.
• Radius: The distance from the center to any point on the circle.
• Diameter: A line segment that passes through the center and has its endpoints on the circle (Diameter = 2 * Radius).*

Standard Form of a Circle

• The standard form of a circle's equation is:

  (x - h)² + (y - k)² = r²

  where

  • (h, k) is the center of the circle
  • r is the radius.

Example of Finding the Center and Radius

• Given the equation: (x - 3)² + (y - 4)² = 3²
  • Center: (h, k) = (3, 4)
  • Radius: r = 3

General Form of a Circle

• The general form of a circle's equation is:

  x² + y² + dx + ey + f = 0

  • To find the center using general form, use the shortcut formulas:
    • Center x-coordinate: -d/2
    • Center y-coordinate: -e/2

Transforming General Form to Standard Form

• Steps to convert general form to standard form:
  1. Rearrange the equation by grouping x and y terms.
  2. Complete the square for both x and y.
  3. Rewrite as a perfect square trinomial and simplify.

Example Problem 1

• Given: x² + y² - 10x + 4y - 7 = 0
  • Rearranged: x² - 10x + y² + 4y = 7
  • Completing the square yields:
    • Center: (5, -2)
    • Radius: 6

Example Problem 2

• Given: x² + y² + 2x - 6y - 15 = 0
  • Rearranged: x² + 2x + y² - 6y = 15
  • Completing the square yields:
    • Center: (-1, 3)
    • Radius: 5

Transforming Standard Form to General Form

• Steps to convert standard form to general form:
  1. Expand the squared terms.
  2. Rearrange the equation into general form.

Example Problem 1

• Given: (x + 5)² + (y - 3)² = 49

  • Expanded:

  x² + 10x + 25 + y² - 6y + 9 = 49

  • Rearranged: x² + y² + 10x - 6y - 15 = 0

Finding Equation Given Center and Radius

• To find the standard form given center and radius, use:

  • Center (h, k): Substitute into standard form.
  • Example: Center = (-4, 3) and radius = √5, results in

  (x + 4)² + (y - 3)² = 5

Graphing the Circle

• To graph a circle:
  1. Identify the center on the Cartesian plane.
  2. Use the radius to determine points around the center.
  3. Draw the circle connecting these points.

Example Problem: Graphing a Circle

• General form: 4x² + 4y² + 8x - 16y - 80 = 0
  • Standard form and center found as part of the solution process.

Assignment

• Transform the general form of the circle 4x² + 4y² + 12x - 4y - 90 = 0 into standard form, and graph the circle.