Lecture Notes: Equation of a Circle

Definition of a Circle

• A circle is the set of all points equidistant from a fixed point, known as the center.
• The radius is the distance from the center to any point on the circle.

Equations of a Circle

• General Form (center at any point (h, k)):
  • ((x - h)^2 + (y - k)^2 = r^2)
• Simplified Form (center at the origin (0, 0)):
  • (x^2 + y^2 = r^2)

Explanation of the Circle Equation

• A right triangle can be formed by:
  • The distance from the origin to ((x, y)) which is (x) units parallel to the x-axis and (y) units parallel to the y-axis.
  • Using the Pythagorean Theorem: (x^2 + y^2 = r^2) describes the relationship.

Examples

• Example 1: Equation for a circle with center (0,0) and radius 3:
  • Formula: (x^2 + y^2 = 9)
• Example 2: Equation with radius (\frac{1}{2}):
  • Formula: (x^2 + y^2 = \frac{1}{4})
• Example 3: Determine the radius from the equation (x^2 + y^2 = 36):
  • Radius (r = 6) units.
• Example 4: Circle centered at origin passing through ((5, 3)):
  • Find (r^2) by substituting (x = 5), (y = 3) into the equation.
  • Result: (x^2 + y^2 = 34).

Determining Point Position Relative to a Circle

• If on the circle: (x^2 + y^2 = r^2)
• If outside the circle: (x^2 + y^2 > r^2)
• If inside the circle: (x^2 + y^2 < r^2)
• Example: Point ((-5, 9)) relative to (x^2 + y^2 = 100)
  • Calculation shows ((-5)^2 + 9^2 = 106) which is greater than 100, thus outside the circle.

Equation of a Circle Not Centered at the Origin

• Given: Center ((3,4)), radius 8
• Equation: ((x - 3)^2 + (y - 4)^2 = 64)

Shortest Distance from a Point to the Circle

• Tip: Shortest distance follows a line through the circle's center.
• Use the distance formula between the point and the center, subtract the radius.
• Example: From point ((10, 7)) to circle (x^2 + y^2 = 49)
  • Full distance: (\sqrt{149})
  • Subtract radius (7), shortest distance (= 5.21) units.

Conclusion

• Practice problems available at Jensen Math for further learning.