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. Recall the standard parametric equations for a circle with center \((h, k)\) and radius \(r\):
2. \[ x = h + r \cos(t) \]
3. \[ y = k + r \sin(t) \]
4. Identify the center \((h, k)\) and radius \(r\) from the problem: center \((3, 5)\) and radius \(6\).
5. Substitute these values into the parametric equations to get:
6. \[ x = 3 + 6 \cos(t) \]
7. \[ y = 5 + 6 \sin(t) \]

Key Concepts

Parametric Equations of a Circle

Parametric equations represent a circle by expressing x and y coordinates as functions of a parameter, usually t. For a circle centered at (h, k) with radius r, the equations are x = h + r cos(t) and y = k + r sin(t), where t varies from 0 to 2π.

Understanding the Center and Radius

The center (h, k) of a circle determines its position on the coordinate plane, while the radius r defines its size. These values are essential for constructing the parametric equations, as they shift and scale the standard unit circle equations accordingly.

Parameter Range and Interpretation

The parameter t typically represents an angle in radians and varies from 0 to 2π to trace the entire circle once. Understanding this range helps in visualizing the motion along the circle and ensures the parametric equations cover all points on the circle.