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

Answer: x^2 + y^2 = 25

Question 2: Calculate the radius of a circle with equation x^2 + y^2 = 49.

Answer: 7

Question 3: Graph the circle given by x^2 + y^2 = 16 on the grid.

Answer: Radius = 4

Question 4: Identify the mistake in the equation: x^2 + y^2 = -25.

Answer: Radius cannot be negative; equation is invalid for a circle

Question 1: A circle has the equation x^2 + y^2 = 36. If a point (6, 0) is on this circle, explain whether this point lies on the circle and justify your answer.

Answer: Yes, because substituting x=6, y=0 gives 36, satisfying the equation.

Question 2: A student writes the equation of a circle as x^2 + y^2 = 49. They then draw it and notice it doesn't pass through (0, 7). Identify and explain the mistake.

Answer: The equation is correct; (0,7) satisfies it, so the circle passes through (0,7). No mistake.

Question 3: Given the equation x^2 + y^2 = r^2, derive the value of r if the circle passes through (3, 4).

Answer: r = 5

Question 1: A circular fountain has a radius of 10 meters. Write its equation assuming it is centered at the origin, and explain how you would graph it.

Answer: Equation: x^2 + y^2 = 100. To graph, plot points at a radius of 10 units from the origin in all directions.

Question 2: A drone flies in a circle with the equation x^2 + y^2 = 25. If it starts at (5, 0), describe its path and how you would verify its position after some time.

Answer: Path is a circle of radius 5 centered at origin. Verify position by measuring distance from origin or plotting points on the grid.

Question 1: If the equation of a circle is x^2 + y^2 = 81, find the length of the diameter and explain your reasoning.

Answer: Diameter = 2 × radius = 2 × 9 = 18

Question 2: A circle's equation is given as x^2 + y^2 = r^2, but the radius is unknown. The circle passes through (0, -12). Find r and write the complete equation.

Answer: r = 12; Equation: x^2 + y^2 = 144

Question 1: A student writes the equation y^2 + x^2 = 100, claiming the order of variables is important. Is this correct? Justify your answer.

Answer: No, because addition is commutative; the order of x^2 and y^2 does not matter.

Question 2: A wrong circle equation is given as y^2 + x^2 = -36. Explain the mistake and how to correct it.

Answer: Radius cannot be negative; the equation is invalid for a circle.

Question 3: Rewrite the following incorrect equation into the standard form: y^2 + x^2 = 49.

Answer: x^2 + y^2 = 49