Where:

• Center: $(h, k)$
• Radius: $r$

Example

Circle with center $(3, -2)$ and radius $5$: $(x - 3)^2 + (y + 2)^2 = 25$

General Form

$x^2 + y^2 + Dx + Ey + F = 0$

To convert to standard form:

1. Group $x$ terms and $y$ terms
2. Complete the square for both
3. Identify center and radius

Finding Center and Radius

From: $x^2 + y^2 - 6x + 4y - 12 = 0$

Step 1: Group and move constant $(x^2 - 6x) + (y^2 + 4y) = 12$

Step 2: Complete the square $(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4$

Step 3: Factor $(x - 3)^2 + (y + 2)^2 = 25$

Center: $(3, -2)$, Radius: $5$

Graphing

1. Plot the center point
2. Count $r$ units in all directions
3. Sketch the circle

Center: $(h, k) = (0, 0)$ Radius: $r = 4$

$(x - 0)^2 + (y - 0)^2 = 4^2$

Simplify: $x^2 + y^2 = 16$

Answer: $x^2 + y^2 = 16$

Rewrite as: $(x - (-1))^2 + (y - 4)^2 = 6^2$

Center: $(-1, 4)$

Radius: $r = \sqrt{36} = 6$

Answer: Center $(-1, 4)$, radius $6$

Step 2: Complete the square

• For $x$: $(8/2)^2 = 16$
• For $y$: $(-2/2)^2 = 1$

$(x^2 + 8x + 16) + (y^2 - 2y + 1) = -8 + 16 + 1$

Step 3: Factor and simplify $(x + 4)^2 + (y - 1)^2 = 9$

Answer:

• Standard form: $(x + 4)^2 + (y - 1)^2 = 9$
• Center: $(-4, 1)$
• Radius: $3$