Question 1: Calculate the radius r of a circle with equation x² + y² = 25.

Answer: 5

Question 2: Write the equation of a circle with radius 7 centered at the origin.

Answer: x² + y² = 49

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

Answer: Student draws a semicircle with radius 5

Question 1: A circle has the equation x² + y² = r². If the circle passes through the point (3, 4), find the value of r. Explain your steps.

Answer: r = 5

Question 2: A student mistakenly writes the equation of a circle as x² + y² = 16 + 4x. Identify the mistake and correct the equation to represent a circle centered at (−2, 0).

Answer: Correct equation: (x + 2)² + y² = 20

Question 1: A radar station detects an aircraft flying along a circular path centered at the station. The equation of the circle is x² + y² = 36, where the radar station is at the origin. If the aircraft is at point (6, 0), how far has it traveled along the circular path from the point (6, 0) to a point directly above the station? Assume the path is a perfect circle.

Answer: π/2 * radius = π/2 * 6 = 3π units

Question 1: Given the circle x² + y² = 64, find the coordinates of the points where the circle intersects the line y = 3x + 4. Show all your working.

Answer: Solutions: (x, y) = (−4, −4), (4, 16)

Question 2: Derive the general form of the circle equation with radius r and center at (h, k). Then, explain how shifting the circle affects the equation.

Answer: General form: (x−h)² + (y−k)² = r². Shifting the circle horizontally or vertically changes h and k accordingly.

Question 1: A student states that the equation y² + x² = 36 represents a circle with radius 6 centered at the origin. Is this correct? If not, identify and correct the mistake.

Answer: Incorrect: the equation is symmetric but the order of terms doesn't matter; the radius is √36=6, the equation is correct.

Question 2: A student writes the equation as x² + y² = 25 + 4x. Is this the correct form of a circle? If not, fix the mistake.

Answer: Correct form: (x + 2)² + y² = 29