Question 1: Write the equation of a circle with center at (3, -2) and radius 5.

Answer: (x - 3)^2 + (y + 2)^2 = 25

Question 2: Calculate the radius of a circle whose equation is x^2 + y^2 - 6x + 8y + 9 = 0.

Answer: 5

Question 3: Determine the coordinates of the center of the circle x^2 + y^2 + 4x - 10y + 12 = 0.

Answer: (-2, 5)

Question 4: Construct the circle with center at (0, 0) and radius 4 on the grid. Describe the steps.

Answer: Plot point (0,0). From this point, measure 4 units in all directions to draw the circle.

Question 1: A circle passes through points A(1, 2), B(5, 6), and C(3, 8). Find the coordinates of its center and radius.

Answer: Center approximately at (3, 5), radius approximately 3.16

Question 2: Explain why the equation x^2 + y^2 + 6x - 8y + 9 = 0 does not represent a real circle.

Answer: Because the discriminant of the equation is negative, indicating no real solutions.

Question 3: A circle is centered at (2, -3) and passes through (5, 1). Find its equation.

Answer: (x - 2)^2 + (y + 3)^2 = 25

Question 4: Prove that the locus of points equidistant from two fixed points, A and B, is the perpendicular bisector of segment AB.

Answer: Any point equidistant from A and B satisfies the equation of the perpendicular bisector, as the distances are equal by definition.

Question 1: A circular swimming pool has a radius of 10 meters. A safety rope is tied at the center and extends to the edge. How long is the rope?

Answer: 10 meters

Question 2: A drone is programmed to fly in a circular path around a fixed point at a radius of 50 meters. If the drone completes one full circle, what is the length of its path?

Answer: approximately 314.16 meters

Question 1: Given the circle with equation x^2 + y^2 - 4x + 6y + 9 = 0, find the coordinates of the center and verify if the point (2, -3) lies on the circle.

Answer: Center at (2, -3), point lies on the circle since it satisfies the equation.

Question 2: Construct the circle with center at (-4, 1) passing through the point (0, 5). Then, determine the equation of the circle.

Answer: Radius is 5; equation is (x + 4)^2 + (y - 1)^2 = 25.

Question 3: A circle passes through three points: P(1, 2), Q(4, 6), R(5, 2). Find the equation of the circle and confirm that all three points lie on it.

Answer: Equation approximately (x - 3)^2 + (y - 4)^2 = 10, all points satisfy the equation.

Question 1: A student writes the equation of a circle as x^2 + y^2 + 8x + 6y + 9 = 0. Identify and correct the mistake to find the proper circle equation.

Answer: The constant term should be adjusted after completing the square; corrected equation: (x + 4)^2 + (y + 3)^2 = 16.

Question 2: A circle with center at (0,0) and radius 7 is moved 3 units right and 4 units down. Write the new equation.

Answer: (x - 3)^2 + (y + 4)^2 = 49