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

Answer: x² + y² = 25

Question 2: Calculate the radius of a circle given the equation x² + y² = 36.

Answer: 6

Question 3: Plot the graph of y = √(25 - x²).

Answer: Plot of the upper semi-circle with radius 5 centered at the origin

Question 4: Construct a circle with radius 7 centered at point (3, 4). Write its equation.

Answer: (x - 3)² + (y - 4)² = 49

Question 5: The equation of a circle is x² + y² = 20. Find the length of its diameter.

Answer: 8√5

Question 6: Explain why the equation x² + y² = r² always represents a circle centered at the origin.

Answer: Because the equation describes all points (x,y) at a fixed distance r from the origin, forming a circle.

Question 7: Given the equation x² + y² = 49, what are the coordinates of the points where the circle intersects the x-axis?

Answer: (-7, 0) and (7, 0)

Question 8: A circle with radius 4 is translated 3 units right and 2 units up. Write its new equation.

Answer: (x - 3)² + (y - 2)² = 16

Question 9: Determine the equation of a circle with center at (−2, 3) and passing through (1, 7).

Answer: (x + 2)² + (y - 3)² = 25

Question 10: Identify the common mistake in the following incorrect circle equation: x² + y² = −16. Why is this invalid?

Answer: Because a radius cannot be negative; r² must be positive, so the equation is invalid as written.

Question 11: Challenge: Find the equation of a circle passing through points (0, 0), (4, 0), and (0, 3).

Answer: (x - 2)² + (y - 1.5)² = 6.25