Question 1: Write the equation of a circle with radius 5 centered at the origin.

Answer: x² + y² = 25

Question 2: Given the circle equation x² + y² = 36, what is its radius?

Answer: 6

Question 3: Plot the graph of y = √(25 - x²) on the grid. What is the radius of the circle?

Answer: 5

Question 4: Construct a circle with a radius of 8. Write its equation centered at (0,0).

Answer: x² + y² = 64

Question 5: If a circle has the equation x² + y² = r² and passes through the point (3,4), what is the radius?

Answer: 5

Question 6: Calculate the radius of a circle with the equation x² + y² = 49.

Answer: 7

Question 7: A circle equation is given as x² + y² = r². If the circle passes through the point (-6, 8), determine the value of r².

Answer: 100

Question 8: Error analysis: A student wrote the equation x² + y² = 36 as x² + y² = 6. Identify and correct the mistake.

Answer: The correct equation should be x² + y² = 36 because the radius squared is 36, not 6.

Question 9: Problem Solving: A circle has an equation x² + y² = r². The circle passes through (0, -7) and (7, 0). Find the radius r.

Answer: 7

Question 10: Real-world application: A circular pond is modeled by the equation x² + y² = 100. If the pond’s center is at (3, -4), write the equation of the pond’s boundary.

Answer: (x - 3)² + (y + 4)² = 100

Question 11: Challenge & Extension: Derive the equation of a circle with radius 10 and center at (4, -3).

Answer: (x - 4)² + (y + 3)² = 100

Question 12: Given the equation x² + y² = 81, identify the error if a student attempts to plot a circle with radius 9 centered at (2, 3). Is the point (2, 12) on the circle? Explain your reasoning.

Answer: No, (2, 12) is not on the circle because plugging in y=12 gives x² + (12)² = 81 → x² + 144 = 81, which is false. Also, the radius is 9, but the point (2,12) is 9 units above (2,3), so it is at a distance 9 from (2,3), but not from the origin. The correct check involves the distance from the center if considering a shifted circle.