The Main Idea

• Circle Definition:
  • Set of points equidistant from a center point in a plane
  • Center: [latex](h,k)[/latex]
  • Radius: [latex]r[/latex]
• Standard Form Equation:
  • [latex](x-h)^2+(y-k)^2 = r^2[/latex]
  • [latex](h,k)[/latex] represents the center
  • [latex]r[/latex] is the radius
• Components of the Equation:
  • [latex](x-h)^2[/latex]: horizontal distance from center
  • [latex](y-k)^2[/latex]: vertical distance from center
  • Right side [latex]r^2[/latex]: squared radius
• Variations:
  • Center at origin: [latex]x^2+y^2 = r^2[/latex]
  • Negative values inside parentheses change to addition

The Main Idea

• Equation to Graph Relationship:
  • Standard form: [latex](x-h)^2 + (y-k)^2 = r^2[/latex]
  • Each part of the equation corresponds to a graphical feature
• Key Components for Graphing:
  • Center: [latex](h,k)[/latex]
  • Radius: [latex]r[/latex] (square root of the right side)
• Critical Points:
  • Center: [latex](h,k)[/latex]
  • Top: [latex](h, k+r)[/latex]
  • Bottom: [latex](h, k-r)[/latex]
  • Right: [latex](h+r, k)[/latex]
  • Left: [latex](h-r, k)[/latex]
• Symmetry:
  • Circles are symmetrical about their center
  • Vertical and horizontal lines through the center are lines of symmetry

The Main Idea

• General Form Equation:
  • [latex]x^2 + y^2 + ax + by + c = 0[/latex]
  • [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are constants
  • Does not directly show center or radius
• Comparison with Standard Form:
  • Standard Form: [latex](x-h)^2+(y-k)^2 = r^2[/latex]
  • Center: [latex](h, k)[/latex]
  • Radius: [latex]r[/latex]
• Conversion Process:
  • Use completing the square for both x and y terms
  • Transforms general form to standard form
  • Reveals center and radius
• Importance of Conversion:
  • Makes graphing easier
  • Helps in identifying key circle properties

The Main Idea

• Standard Form Equation: [latex](x - h)^2 + (y - k)^2 = r^2[/latex]
  • [latex](h, k)[/latex] is the center
  • [latex]r[/latex] is the radius
• Three Common Scenarios:
  • Known center and radius
  • Known center and a point on the circle
  • Known two points on the circle (often diameter endpoints)
• Key Formulas:
  • Distance Formula: [latex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/latex]
  • Midpoint Formula: [latex]\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)[/latex]

Scenario Breakdown

Scenario 1: Known Center and Radius

• Directly plug into standard form equation

Scenario 2: Known Center and Point

1. Use distance formula to find radius
2. Plug center and calculated radius into standard form

Scenario 3: Two Points on Circle

1. Use midpoint formula to find center
2. Use distance formula to find radius (center to either point)
3. Plug center and calculated radius into standard form

The Main Idea

• System of Equations:
  • Circle: [latex](x - h)^2 + (y - k)^2 = r^2[/latex]
  • Line: [latex]y = mx + b[/latex]
• Solving Process:
  • Substitute line equation into circle equation
  • Solve resulting quadratic equation
• Types of Intersections:
  • No intersection: No real solutions
  • Tangent: One real solution
  • Two intersections: Two real solutions
• Geometric Interpretation:
  • No intersection: Line outside circle
  • One intersection: Line touches circle at one point
  • Two intersections: Line passes through circle

Solution Process

1. Substitute [latex]y = mx + b[/latex] into circle equation
2. Simplify to get quadratic in [latex]x[/latex]: [latex]ax^2 + bx + c = 0[/latex]
3. Solve quadratic using preferred method (factoring, quadratic formula, etc.)
4. Find corresponding [latex]y[/latex]-values using line equation
5. Check solutions in both original equations