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Question bank: Analytic Geometry: Equation of the Circle

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Question 1:

Easy

Given the equation of the circle (x - 3)^2 + (y + 4)^2 = 25, identify the center and the radius of the circle. Then, consider a point P(x, y) located on the circle and determine if P is in the interior, on the boundary, or in the exterior of the circle. Use the point's information to check if the provided coordinates correspond to a real solution of the circle equation.

Analytic Geometry: Equation of the Circle

Contextualized Question

Calculus Question

Question 2:

Easy

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.

Analytic Geometry: Equation of the Circle

Contextualized Question

Calculus Question

Question 3:

Hard

Consider the point P located in the first quadrant of the Cartesian plane and equidistant from points A(3, 4) and B(7, 1). Let 'r' be the radius of a circle with center at A passing through P. Determine the equation of the circle in terms of 'r'. Consider 'pi' as an approximation of 3.14 to calculate the numerical value of the final equation.

Analytic Geometry: Equation of the Circle

Contextualized Question

Calculus Question

Question 4:

Hard

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?

Analytic Geometry: Equation of the Circle

Multidisciplinary Question

Question 5:

Hard

Given a circle with center at the origin of the Cartesian plane and passing through the point (6,8), determine the equation of the circle and graphically represent it. Then, find the area of the region of the plane bounded by this circle and the x-axis.

Analytic Geometry: Equation of the Circle

Theorical Question

Contextualized Question

IARA TIP

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