Lecture Notes: Equation of a Circle with Diameter Endpoints

Key Concepts

• Equation of a Circle:
  • Formula: ( (x - h)^2 + (y - k)^2 = r^2 )
  • ( h, k ) represents the center coordinates (( h = x ) value, ( k = y ) value)
  • ( r ) is the radius of the circle

Problem Overview

• Given endpoints of a diameter: (2, 3) and (-4, -5)
• Goal: Find the equation of the circle

Solution Steps

1. Find the Center of the Circle

• Midpoint Formula: Used to find the center of the circle
  • Formula: ( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) )
• Calculation:
  • Midpoint ( = \left( \frac{2 + (-4)}{2}, \frac{3 + (-5)}{2} \right) )
  • ( = \left( \frac{-2}{2}, \frac{-2}{2} \right) )
  • ( = (-1, -1) )
• Result: Center ( (h, k) = (-1, -1) )

2. Determine the Radius

• Radius Options:
  • Distance from center to a point on the circle
  • Half the length of the diameter
• Distance Formula: Used for calculating distance between the center and one endpoint
  • Formula: ( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )
• Calculation:
  • Endpoints: Center ((-1, -1)) and a point (2, 3)
  • ( = \sqrt{(2 - (-1))^2 + (3 - (-1))^2} )
  • ( = \sqrt{(3)^2 + (4)^2} )
  • ( = \sqrt{9 + 16} )
  • ( = \sqrt{25} )
  • ( = 5 )
• Result: Radius ( r = 5 )

3. Write the Equation of the Circle

• Substitute ( h, k, ) and ( r ) into the circle equation
• Simplification Steps:
  • ((x - (-1))^2 + (y - (-1))^2 = 5^2)
  • ((x + 1)^2 + (y + 1)^2 = 25)
• Final Equation: ((x + 1)^2 + (y + 1)^2 = 25)

Conclusion

• Equation of the circle: ((x + 1)^2 + (y + 1)^2 = 25)
• Center: ((-1, -1))
• Radius: 5