Circle Equations

Grade 10 · geometry · 51 practice problems · read aloud

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Circle Equations: Your Complete Guide

1. What is a Circle Equation? 🎯

A circle equation is a mathematical formula that describes all points equidistant from a center point. The standard form is (x - h)² + (y - k)² = r², where (h,k) is the center and r is the radius.

Why it's useful: It helps us graph circles, solve geometry problems, and model real-world situations like satellite orbits or wheel rotations.

2. Step-by-Step Guide

  1. Identify the center (h,k) from the equation
  2. Find the radius r by taking the square root of the right side
  3. Plot the center on your coordinate plane
  4. Mark points r units up, down, left, and right from center
  5. Connect these points with a smooth curve

3. Worked Examples

Example 1: (x - 2)² + (y + 1)² = 9

Step 1: Center is (2, -1) → watch the signs!

Step 2: Radius = √9 = 3

Step 3: Plot center at (2, -1), then points 3 units away in all directions.

Example 2: x² + y² = 16

Step 1: Center is (0, 0) → no h or k values shown

Step 2: Radius = √16 = 4

Step 3: Plot center at origin, mark points 4 units away.

4. Common Mistakes ⚠️

Sign errors: (x - 3)² means h = 3, but (x + 3)² means h = -3

Radius confusion: If r² = 25, then r = 5 (not 25!)

Forgetting to square root: Always take √ of the constant term

Center mix-up: (h,k) is the center, not (x,y)

5. Tips & Tricks

Memory aid: "Happy King" for (h,k) coordinates

Quick check: The right side must always be positive

Shortcut: If you see x² + y², center is automatically (0,0)

Verification: Plug center coordinates back in - they should make the equation = 0

6. Practice Suggestions

  • Start with simple circles centered at (0,0)
  • Practice identifying centers and radii from equations
  • Work backwards: given center and radius, write the equation
  • Try real applications: "A circular garden has center at (5,2) and radius 8m - write its equation"
  • Use graph paper to plot multiple circles and see their relationships

Remember: Circle equations follow consistent patterns. With practice, you'll recognize them instantly! 📐

Practice problems

6 of the 51, worked through step by step — try them before opening the answer.

1 (x - 5)² + (y + 3)² = 49

Hint: Compare the given equation to the standard form of a circle equation to identify the center coordinates and radius.

Show the answer

Answer: Center: (5, -3), Radius: 7

  1. The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
  2. Compare (x - 5)² + (y + 3)² = 49 with the standard form.
  3. Identify h = 5, k = -3 (since y + 3 = y - (-3)), and r² = 49.
  4. Calculate r = sqrt(49) = 7.
  5. The center is (5, -3) and the radius is 7.

2 (x + 5)² + (y - 3)² = 49

Hint: Compare the given equation to the standard form of a circle equation to identify the center coordinates and radius.

Show the answer

Answer: (-5, 3), 7

  1. The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
  2. Compare (x + 5)² + (y - 3)² = 49 to the standard form.
  3. (x + 5)² can be written as (x - (-5))², so h = -5.
  4. (y - 3)² gives k = 3.
  5. The right side is 49, which equals r², so r = √49 = 7.
  6. Therefore, the center is (-5, 3) and the radius is 7.

3 (x - 6)² + (y + 1)² = 16

Hint: Compare the given equation to the standard form of a circle equation to identify the center coordinates and radius.

Show the answer

Answer: Center: (6, -1), Radius: 4

  1. The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
  2. Compare (x - 6)² + (y + 1)² = 16 to the standard form.
  3. Identify h = 6, k = -1 (since y + 1 = y - (-1)), and r² = 16.
  4. Calculate the radius: r = √16 = 4.
  5. The center is (6, -1) and the radius is 4.

4 (x - 4)² + (y + 6)² = 64

Hint: Compare the given equation to the standard form of a circle equation to identify the center coordinates and radius.

Show the answer

Answer: Center: (4, -6), Radius: 8

  1. The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
  2. Compare (x - 4)² + (y + 6)² = 64 with the standard form.
  3. Identify h = 4, k = -6 (since y + 6 = y - (-6)), and r² = 64.
  4. Calculate r = √64 = 8.
  5. Therefore, the center is (4, -6) and the radius is 8.

5 (x - 1)² + (y + 6)² = 16

Hint: Compare the given equation to the standard form of a circle equation to identify the center coordinates and radius.

Show the answer

Answer: Center: (1, -6), Radius: 4

  1. The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h,k) is the center and r is the radius.
  2. Compare (x - 1)² + (y + 6)² = 16 to the standard form.
  3. The x-coordinate of the center is h = 1 (from x - 1).
  4. The y-coordinate of the center is k = -6 (from y + 6 which is y - (-6)).
  5. The radius squared is r² = 16, so r = √16 = 4.
  6. Therefore, the center is (1, -6) and the radius is 4.

6 (x - 7)² + (y + 5)² = 81

Hint: Compare the given equation to the standard form of a circle equation to identify the center coordinates and radius.

Show the answer

Answer: Center: (7, -5), Radius: 9

  1. The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h,k) is the center and r is the radius.
  2. Compare (x - 7)² + (y + 5)² = 81 to the standard form.
  3. The x-coordinate of the center is h = 7 (since x - 7 means h = 7).
  4. The y-coordinate of the center is k = -5 (since y + 5 means y - (-5), so k = -5).
  5. The radius squared is r² = 81, so r = √81 = 9.
  6. Therefore, the center is (7, -5) and the radius is 9.
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