Question bank: Analytic Geometry: Equation of the Circle

Consider a circle C with equation (x-3)² + (y+2)² = 25. A point P moves in such a way that the sum of the squares of its distances to the intersection points of C with the x-axis is constant and equal to 50. Determine the initial position of P on the Cartesian plane.

In an amusement park, there is a Ferris wheel located at a point with known coordinates representing the center of the circumference traveled by its cabins. Two lights, one at each horizontal end of the wheel's diameter, project onto the ground forming a 60-degree angle with the line connecting the center of the wheel to the projection points. A visitor observes that the intersection point of these lights is at a distance of 10 meters from the center of the Ferris wheel. (a) Considering that the center of the Ferris wheel is the point (h, k) and the radius is R, determine the equation of the circumference that describes the movement of the Ferris wheel cabins. (b) Evaluate and discuss the visitor's position in relation to the equation and the radius of the Ferris wheel, providing a mathematical justification for your conclusion.

An architect is designing a new building in a city where urban space is a scarce and expensive resource. To maximize land use, he decides that the shape of the building's base will be a circle. The architect wants the circle to have its center at point C(2, 5) and for a point A(4, 7) on the edge of the circle to also be one of the base points of the building. Additionally, the architect decides that the circle should pass through a point B(6, 1) which is located at a maximum distance of 6 units from the center of the circle. Considering this information, calculate the equation of the circle that will represent the building's base. Explain, in terms of analytic geometry, how the architect's decision to use a circle as the building's base can affect the design and structure of the building, considering also aspects of engineering and architectural design.

Consider a circle C in the Cartesian plane defined by the equation (x−h)² + (y−k)² = r², where (h, k) is the center of the circle and r is the radius. An observer is positioned at point P(-3, 4) and sights the tangent T to circle C at the point of tangency Q(-1, 2). Let A be the orthogonal projection of P onto tangent T and B the orthogonal projection of P onto the line containing the diameter of the circle and point Q. What is the area of the quadrilateral PABQ, in units of the Cartesian plane area?

Consider a Cartesian map where the coordinates represent geographical position and a reference point A, with coordinates (3, -2), marking the center of a circular park. The organizers of an annual event in the park wish to build a stage exactly on the edge of the circle, which will have a radius of 10 units. Determine the equation of the circumference representing the park's edge, considering that the coordinates (x, y) of the point on the circle are given in meters and that point A is the center of the circle. Additionally, identify the coordinates of the point on the circle closest to point B, with coordinates (9, 7), where the stage should be built. Explain the steps to find the solution and justify your answer.