Circle Equations

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Answer Key & Explanations

Circle Equations · Grade 10 · Worksheet 2

1. (x + 2)² + (y - 7)² = 121 Answer: Center: (-2, 7), Radius: 11 Solution: 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. Compare (x + 2)² + (y - 7)² = 121 to the standard form.
   Full step-by-step solution

   Step 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. Step 2: Compare (x + 2)² + (y - 7)² = 121 to the standard form. Step 3: (x + 2)² can be written as (x - (-2))², so h = -2. Step 4: (y - 7)² shows k = 7. Step 5: The right side is 121, which equals r², so r = √121 = 11. Step 6: Therefore, the center is (-2, 7) and the radius is 11.

2. (x - 6)² + (y + 8)² = 100 Answer: Center: (6, -8), Radius: 10 Solution: 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. Compare (x - 6)² + (y + 8)² = 100 with the standard form. For the x-term: (x - 6)² means h = 6.
   Full step-by-step solution

   Step 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. Step 2: Compare (x - 6)² + (y + 8)² = 100 with the standard form. Step 3: For the x-term: (x - 6)² means h = 6. Step 4: For the y-term: (y + 8)² can be written as (y - (-8))², so k = -8. Step 5: The right side is 100, which equals r², so r² = 100. Step 6: Take the square root of both sides: r = √100 = 10. Step 7: Therefore, the center is (6, -8) and the radius is 10.

3. Matiu is analyzing a circle drawn on a coordinate plane. The circle passes through the points (4, 0), (0, 4), and (-4, 0). What is the radius of this circle? Answer: 4 Solution: Observe the given points: (4, 0), (0, 4), and (-4, 0). Notice the symmetry. The points (4, 0) and (-4, 0) are symmetric across the y-axis, and (0, 4) is on the y-axis.
   Full step-by-step solution

   Step 1: Observe the given points: (4, 0), (0, 4), and (-4, 0). Step 2: Notice the symmetry. The points (4, 0) and (-4, 0) are symmetric across the y-axis, and (0, 4) is on the y-axis. This suggests the center is at (0, 0). Step 3: To confirm, check if the distance from (0, 0) to each point is the same. Step 4: Distance from (0, 0) to (4, 0) = sqrt((4-0)^2 + (0-0)^2) = sqrt(16) = 4. Step 5: Distance from (0, 0) to (0, 4) = sqrt((0-0)^2 + (4-0)^2) = sqrt(16) = 4. Step 6: Distance from (0, 0) to (-4, 0) = sqrt((-4-0)^2 + (0-0)^2) = sqrt(16) = 4. Step 7: Since all distances are equal, the center is (0, 0) and the radius is 4. The answer is 4.

4. Matiu is planning a circular garden for his school. The garden's boundary follows the equation x² + y² - 14x + 8y - 56 = 0. What is the radius of Matiu's garden in meters? Answer: D. 11 m Solution: The general form of a circle equation x² + y² + Dx + Ey + F = 0 can be converted to standard form (x-h)² + (y-k)² = r² by completing the square.
   Full step-by-step solution

   The general form of a circle equation x² + y² + Dx + Ey + F = 0 can be converted to standard form (x-h)² + (y-k)² = r² by completing the square. This process involves rearranging the x and y terms, adding appropriate constants to both sides to create perfect square trinomials, and then simplifying to identify the center coordinates and radius.

5. (x - 11)² + (y + 9)² = 144 Answer: Center: (11, -9), Radius: 12 Solution: 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. Compare (x - 11)² + (y + 9)² = 144 with the standard form. The x-coordinate of the center is h = 11.
   Full step-by-step solution

   Step 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. Step 2: Compare (x - 11)² + (y + 9)² = 144 with the standard form. Step 3: The x-coordinate of the center is h = 11. Step 4: The y-coordinate is k = -9 (since y + 9 = y - (-9)). Step 5: The radius squared is r² = 144, so r = √144 = 12. Step 6: Therefore, the center is (11, -9) and the radius is 12.

6. Find the center and radius of the circle: (x - 5)² + (y + 7)² = 81 Answer: Center: (5, -7), Radius: 9 Solution: 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. Compare (x - 5)² + (y + 7)² = 81 with the standard form. The x-coordinate of the center is h = 5.
   Full step-by-step solution

   Step 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. Step 2: Compare (x - 5)² + (y + 7)² = 81 with the standard form. Step 3: The x-coordinate of the center is h = 5. Step 4: The y-coordinate of the center is k = -7 (since y + 7 = y - (-7)). Step 5: The radius squared is r² = 81. Step 6: Take the square root of both sides: r = √81 = 9. Step 7: Therefore, the center is (5, -7) and the radius is 9.

7. Find the center and radius of the circle: (x - 11)² + (y + 14)² = 169 Answer: Center: (11, -14), Radius: 13 Solution: 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. Compare the given equation (x - 11)² + (y + 14)² = 169 to the standard form.
   Full step-by-step solution

   Step 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. Step 2: Compare the given equation (x - 11)² + (y + 14)² = 169 to the standard form. Step 3: Identify h = 11 (since x - 11 means h = 11). Step 4: Identify k = -14 (since y + 14 means y - (-14), so k = -14). Step 5: The center is (11, -14). Step 6: The right side is 169, which equals r², so r² = 169. Step 7: Take the square root of both sides: r = √169 = 13. Step 8: The radius is 13. The center is (11, -14) and the radius is 13.

8. Matiu is designing a circular garden for his school project. He needs to find the center and radius of a circle represented by the equation x² + y² - 14x + 8y - 20 = 0. Which of the following correctly identifies the center and radius of Matiu's circle? Answer: A. Center: (7, -4), Radius: √85 Solution: Start with the given equation: x² + y² - 14x + 8y - 20 = 0 Group x terms and y terms: (x² - 14x) + (y² + 8y) = 20 Complete the square for x terms: x² - 14x + 49 = (x - 7)² Complete the square for y terms: y² + 8y + 16 = (y + 4)² Add the constants to both sides: (x² - 14x + 49) + (y² + 8y + 16) =…
   Full step-by-step solution

   Step 1: Start with the given equation: x² + y² - 14x + 8y - 20 = 0 Step 2: Group x terms and y terms: (x² - 14x) + (y² + 8y) = 20 Step 3: Complete the square for x terms: x² - 14x + 49 = (x - 7)² Step 4: Complete the square for y terms: y² + 8y + 16 = (y + 4)² Step 5: Add the constants to both sides: (x² - 14x + 49) + (y² + 8y + 16) = 20 + 49 + 16 Step 6: Simplify: (x - 7)² + (y + 4)² = 85 Step 7: The center is (7, -4) and the radius is √85 The correct answer is Center: (7, -4), Radius: √85.