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

Answer: x^2 + y^2 = 25

Question 2: What is the radius of the circle given by the equation x^2 + y^2 = 64?

Answer: 8

Question 3: Plot the circle defined by x^2 + y^2 = 9 on the grid.

Answer: Circle with radius 3 centered at (0, 0)

Question 4: Determine whether the point (4, 3) lies on the circle x^2 + y^2 = 25.

Answer: Yes

Question 1: A circle has an equation x^2 + y^2 = 36. Find its radius and diameter. Then, plot the circle on the grid.

Answer: Radius 6, diameter 12

Question 2: If a circle with radius 7 is centered at the origin, write its equation and find the coordinates of a point on the circle where x = 0.

Answer: x^2 + y^2 = 49; point (0, 7) or (0, -7)

Question 3: Explain why all points on the circle x^2 + y^2 = r^2 are exactly r units from the origin.

Answer: Because the equation defines all points (x, y) where the distance from (0, 0) is r, by the distance formula √(x^2 + y^2) = r.

Question 1: A circular fountain at the park is centered at the origin with a radius of 10 meters. Write the equation of the fountain's boundary and determine the area of the fountain.

Answer: x^2 + y^2 = 100; Area = π * 10^2 = 100π square meters

Question 2: A drone flies in a circular path centered at the origin with radius 15 meters. Write its flight path equation and calculate the circumference of its path.

Answer: x^2 + y^2 = 225; Circumference = 2π * 15 ≈ 94.2 meters

Question 1: Derive the general equation of a circle with center at the origin given any radius r, and verify with r = 8.

Answer: x^2 + y^2 = r^2; for r=8, x^2 + y^2 = 64

Question 2: Given the equation x^2 + y^2 = 50, find the length of the radius and identify at least two points on the circle.

Answer: Radius √50 ≈ 7.07; points (0, ±√50), (±√50, 0)

Question 1: Plot the circle with equation x^2 + y^2 = 16 and identify its radius.

Answer: Radius 4

Question 2: Write the equation of a circle centered at the origin with radius 3.

Answer: x^2 + y^2 = 9

Question 3: If the point (0, -5) lies on the circle x^2 + y^2 = r^2, find the radius r.

Answer: 5

Question 1: A student writes the equation x^2 + y^2 = -25 for a circle. Explain the mistake and provide the correct form.

Answer: A circle cannot have a negative radius squared; the correct equation is not possible unless the radius squared is positive.

Question 2: A student writes the equation y^2 + x^2 = 36. Identify the mistake if any, and explain.

Answer: No mistake; the equation correctly represents a circle with radius 6 centered at the origin.