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 5.

💡 Show Solution

Step 1: Use standard form centered at origin: x² + y² = r²

Step 2: Substitute r = 5: x² + y² = 5² x² + y² = 25

Answer: x² + y² = 25

2Problem 2easy

❓ 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: $x^2 + y^2 = 16$

3Problem 3easy

❓ Question:

Identify the center and radius: (x - 2)² + (y + 3)² = 16

💡 Show Solution

Step 1: Recall standard form: (x - h)² + (y - k)² = r² where (h, k) is the center and r is the radius

Step 2: Match to given equation: (x - 2)² + (y - (-3))² = 4²

Step 3: Identify values: h = 2, k = -3, r = 4

Answer: Center (2, -3), radius 4

4Problem 4medium

❓ 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 $6$

5Problem 5medium

❓ Question:

Write the equation in standard form: x² + y² - 6x + 4y - 12 = 0

💡 Show Solution

Step 1: Group x and y terms: (x² - 6x) + (y² + 4y) = 12

Step 2: Complete the square for x: x² - 6x → (x - 3)² - 9

Step 3: Complete the square for y: y² + 4y → (y + 2)² - 4

Step 4: Substitute: (x - 3)² - 9 + (y + 2)² - 4 = 12

Step 5: Simplify: (x - 3)² + (y + 2)² = 12 + 9 + 4 (x - 3)² + (y + 2)² = 25

Step 6: Identify: Center (3, -2), radius 5

Answer: (x - 3)² + (y + 2)² = 25

6Problem 6medium

❓ Question:

Find the equation of the circle with center (1, -2) passing through (4, 2).

💡 Show Solution

Step 1: Find the radius using distance formula: r = √[(x₂ - x₁)² + (y₂ - y₁)²] r = √[(4 - 1)² + (2 - (-2))²] r = √[3² + 4²] r = √[9 + 16] r = √25 = 5

Step 2: Write equation in standard form: (x - h)² + (y - k)² = r²

Step 3: Substitute h = 1, k = -2, r = 5: (x - 1)² + (y + 2)² = 25

Step 4: Verify point (4, 2) satisfies equation: (4 - 1)² + (2 + 2)² = 9 + 16 = 25 ✓

Answer: (x - 1)² + (y + 2)² = 25

7Problem 7hard

❓ 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: $3$

8Problem 8hard

❓ Question:

Determine if the point (3, 4) is inside, on, or outside the circle x² + y² = 16.

💡 Show Solution

Step 1: Identify circle properties: Center (0, 0), radius r = 4

Step 2: Calculate distance from center to point: d = √[(3 - 0)² + (4 - 0)²] d = √[9 + 16] d = √25 = 5

Step 3: Compare distance to radius: d = 5, r = 4 Since d > r, point is OUTSIDE

Alternative method: Step 4: Substitute point into equation: 3² + 4² = 9 + 16 = 25

Step 5: Compare to r²: 25 > 16, so point is OUTSIDE

Answer: Outside the circle

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