Equation of a Circle Notes and Worksheets - Lindsay Bowden - Free Printable

Let’s solve each problem step by step.

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Part 1: Determining if a Point is on a Circle

The equation of a circle is:
(x - h)² + (y - k)² = r²
where (h, k) is the center and r is the radius.

To check if a point (x, y) is on the circle, plug in the x and y values into the left side of the equation. If it equals the right side (r²), then the point IS on the circle.

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Example 1: Is (2, 3) on the circle (x-2)² + (y+1)² = 16?

Plug in x=2, y=3:

Left side: (2 - 2)² + (3 + 1)² = (0)² + (4)² = 0 + 16 = 16

Right side: 16

✔ 16 = 16 → YES, the point (2, 3) IS on the circle.

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Example 2: Is (0, 5) on the circle (x+1)² + (y-4)² = 1?

Note: (x+1) means (x - (-1)), so center is at (-1, 4)

Plug in x=0, y=5:

Left side: (0 + 1)² + (5 - 4)² = (1)² + (1)² = 1 + 1 = 2

Right side: 1

✘ 2 ≠ 1 → NO, the point (0, 5) is NOT on the circle.

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Try: Is (5, 2) on the circle (x+1)² + (y-2)² = 36?

Plug in x=5, y=2:

Left side: (5 + 1)² + (2 - 2)² = (6)² + (0)² = 36 + 0 = 36

Right side: 36

✔ 36 = 36 → YES, the point (5, 2) IS on the circle.

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Part 2: Writing the Equation Given the Center and a Point

Steps:
1. Plug center (h, k) and point (x, y) into the formula to find r².
2. Solve for r² (you don’t need r, just r²).
3. Write the full equation using h, k, and r².

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Example 1: Center (0, 0), point (5, -12)

Step 1: Plug into (x - h)² + (y - k)² = r²
→ (5 - 0)² + (-12 - 0)² = r²
→ 25 + 144 = r²
→ r² = 169

Step 2: Write equation with h=0, k=0, r²=169
→ (x - 0)² + (y - 0)² = 169
→ x² + y² = 169

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Example 2: Center (2, -3), point (-4, 5)

Step 1: Plug in x=-4, y=5, h=2, k=-3
→ (-4 - 2)² + (5 - (-3))² = r²
→ (-6)² + (8)² = r²
→ 36 + 64 = r²
→ r² = 100

Step 2: Write equation
→ (x - 2)² + (y - (-3))² = 100
→ (x - 2)² + (y + 3)² = 100

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Try: Center (-5, -7), point (3, 0)

Step 1: Plug in x=3, y=0, h=-5, k=-7
→ (3 - (-5))² + (0 - (-7))² = r²
→ (3 + 5)² + (0 + 7)² = r²
→ (8)² + (7)² = r²
→ 64 + 49 = r²
→ r² = 113

Step 2: Write equation
→ (x - (-5))² + (y - (-7))² = 113
→ (x + 5)² + (y + 7)² = 113

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Final Answer:

Determining if a Point is on a Circle:
- (2, 3) is ON the circle (x-2)² + (y+1)² = 16 → ✔ Yes
- (0, 5) is NOT on the circle (x+1)² + (y-4)² = 1 → ✘ No
- (5, 2) is ON the circle (x+1)² + (y-2)² = 36 → ✔ Yes

Writing the Equation Given Center and Point:
- Center (0,0), point (5,-12): x² + y² = 169
- Center (2,-3), point (-4,5): (x - 2)² + (y + 3)² = 100
- Center (-5,-7), point (3,0): (x + 5)² + (y + 7)² = 113