Q. 4TB - Calculus

Question

Parametric equations for a circle: Find parametric equations whose graph is the circle with radius ρ centered at the point (a, b) in the xy-plane such that the graph is traced counterclockwise k > 0 times on the interval [0, 2π] starting at the point (a + ρ, b).

Step-by-Step Solution

Answer

The parametric equation of the circle is$\left.\begin{array}{r}x=a+\rho \cos t\\ y=b+\rho \sin t\end{array}\right\} 0\le t\le 2\mathrm{\pi }.$

Step 1. Given Information.

It is given that the circle is centered at the the point $\left(a,b\right)$ with radius $\rho$ and the graph is traced counterclockwise $k>0$ times with a starting point $\left(a+\rho ,b\right).$

Step 2. Find the parametric equation for a circle.

It is given that the circle is centered at the point $\left(a,b\right)$ with the radius $\rho .$Thus, the parametric equation of a circle of center and radius is:

$$\begin{gathered} x=a+\rho \cos t \\ y=b+\rho \sin t \end{gathered}$$

Now, the graph is traced counterclockwise $k>0$ times with a starting point $\left(a+\rho ,b\right).$ Thus, the parametric equation of the circle is:

$\left.\begin{array}{r}x=a+\rho \cos t\\ y=b+\rho \sin t\end{array}\right\} 0\le t\le 2\mathrm{\pi }.$