In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.


Circle: Center: (3,5); Radius: 6

Verified step by step guidance

1. Identify the standard form of the parametric equations for a circle centered at (h, k) with radius r: x(t) = h + r * cos(t), y(t) = k + r * sin(t).
2. Substitute the given center (3, 5) into the equations: h = 3 and k = 5.
3. Substitute the given radius 6 into the equations: r = 6.
4. Write the parametric equations using the substituted values: x(t) = 3 + 6 * cos(t), y(t) = 5 + 6 * sin(t).
5. These parametric equations represent the circle with the specified center and radius.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a variable, typically time (t). For a circle, these equations can be derived using trigonometric functions, where x and y coordinates are defined in terms of a parameter, allowing for a more dynamic representation of the shape.

Circle Equation

The standard equation of a circle in a Cartesian coordinate system is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This equation describes all points that are a fixed distance (the radius) from the center, providing a foundational understanding for deriving parametric equations.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are essential for defining circular motion. In the context of a circle, the x-coordinate can be expressed as x = h + r * cos(t) and the y-coordinate as y = k + r * sin(t), where t is the parameter that varies, tracing the circle as it changes.