Circle Equations: Foundations

Standard Equation of a Circle

The standard form equation of a circle provides key information about the circle's center and radius.

Standard Form Equation

[Diagram of circle with center (h,k), radius r, and point (x,y) on circumference]

(x - h)² + (y - k)² = r²

Converting to Standard Form

Sometimes you'll need to convert from general form to standard form by completing the square.

General Form Equation

x² + y² + Dx + Ey + F = 0

Steps to Convert:

1. Group x and y terms:

(x² + Dx) + (y² + Ey) = -F

2. Complete the square for x and y:

(x² + Dx + (D/2)²) + (y² + Ey + (E/2)²) = -F + (D/2)² + (E/2)²

3. Write in standard form:

(x + D/2)² + (y + E/2)² = (D² + E² - 4F)/4

Key Circle Properties from Equations

Equation Form Center Radius
(x-2)² + (y+3)² = 16 (2, -3) 4
x² + y² = 25 (0, 0) 5
(x+1)² + y² = 9 (-1, 0) 3

Strategies for Solving Circle Equation Problems

  1. Identify the form: Is the equation in standard or general form?
  2. Find the center: Look at (h,k) values in standard form (watch the signs!)
  3. Calculate radius: Take square root of the constant term (must be positive)
  4. For general form: Complete the square to convert to standard form
  5. Graph carefully: Plot center first, then mark radius units in all directions

Practice Question

What is the center and radius of the circle represented by the equation x² + y² - 6x + 4y - 3 = 0?

A) Center: (3, -2); Radius: 4
B) Center: (-3, 2); Radius: 4
C) Center: (3, -2); Radius: 16
D) Center: (-6, 4); Radius: √3

Explanation

We need to complete the square to convert from general form to standard form.

1. Rearrange terms:

x² - 6x + y² + 4y = 3

2. Complete the square for x and y:

(x² - 6x + 9) + (y² + 4y + 4) = 3 + 9 + 4

We added (6/2)² = 9 for x and (4/2)² = 4 for y

3. Write as perfect squares:

(x - 3)² + (y + 2)² = 16

4. Identify center and radius:

  • Center: (3, -2) [Note sign change]
  • Radius: √16 = 4

The correct answer is A) Center: (3, -2); Radius: 4.

Common mistakes:

Additional Tips