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

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Recall the standard parametric equations for a circle with center \((h, k)\) and radius \(r\):

\[ x = h + r \cos(t) \]

\[ y = k + r \sin(t) \]

Identify the center \((h, k)\) and radius \(r\) from the problem: center \((3, 5)\) and radius \(6\).

Substitute these values into the parametric equations to get:

\[ x = 3 + 6 \cos(t) \]

\[ y = 5 + 6 \sin(t) \]

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주요 개념

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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.

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