Question

the circle

$(x - 4)^{2}+(y - 2)^{2}=9$

can be drawn with parametric equations. assuming the circle is traced clockwise as the parameter increases and

$x(t)=4 + 3\cos t$,

then

$y(t)=$

Explanation:

Step1: Recall the standard parametric equations for a circle

The standard parametric equations for a circle \((x - a)^{2}+(y - b)^{2}=r^{2}\) are \(x=a + r\cos t\) and \(y=b + r\sin t\) (counter - clockwise). For clockwise, we use \(x=a + r\cos t\) and \(y=b - r\sin t\).
Here \(a = 4\), \(b = 2\) and \(r = 3\).

Step2: Determine \(y(t)\)

Since \(x(t)=4 + 3\cos t\) (matches the \(x\) - part of the clockwise parametric form \(x=a + r\cos t\)), for the \(y\) - part of the clockwise parametric form \(y=b - r\sin t\) with \(b = 2\) and \(r = 3\), we have \(y(t)=2-3\sin t\).

Answer:

\(y(t)=2 - 3\sin t\)