Circles

Equations and graphs of circles

Circles

Standard Form

$(x - h)^2 + (y - k)^2 = r^2$

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:

Group $x$ terms and $y$ terms
Complete the square for both
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

Plot the center point
Count $r$ units in all directions
Sketch the circle

📚 Practice Problems

1Problem 1easy

❓ Question:

Write the equation of a circle with center $(0, 0)$ and radius $4$ .

💡 Show Solution

Use standard form: $(x - h)^2 + (y - k)^2 = r^2$

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

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

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

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

2Problem 2medium

❓ Question:

Find the center and radius of $(x + 1)^2 + (y - 4)^2 = 36$

💡 Show Solution

Compare to standard form: $(x - h)^2 + (y - k)^2 = r^2$

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

Center: $(-1, 4)$

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

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

3Problem 3hard

❓ Question:

Convert to standard form and find the center and radius: $x^2 + y^2 + 8x - 2y + 8 = 0$

💡 Show Solution

Step 1: Group variables $(x^2 + 8x) + (y^2 - 2y) = -8$

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: $3$

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