Chapter 9.4, Problem 12E

To determine

To find: Which set of parametric equation represents the given graph.

Answer to Problem 12E

  $\begin{array}{l}x=2-\cos \left(\theta \right)\\ y=3-\sin \left(\theta \right)\end{array}$

Explanation of Solution

Given information:

Graph of parametric equations.

Concept:

Parametric equations of a Circle with Centre $\left(h,k\right)$ and radius $r$ is

  $\begin{array}{l}x=h+r\cos \left(\theta \right)\\ y=k+r\sin \left(\theta \right)\end{array}$

From the given graph

On observing

We can say that

Centre of circle $=\left(h,k\right)=\left(2,3\right)$

And Diameter of circle = $=4-2=2$

  $\begin{array}{l}\Rightarrow 2r=2\\ \Rightarrow \frac{2r}{2}=\frac{2}{2}\\ \Rightarrow r=1\end{array}$

Therefore,

Radius of Circle $=r=1$

  $\begin{array}{l}x=h+r\cos \left(\theta \right)\\ y=k+r\sin \left(\theta \right)\end{array}$

On substituting values

We get

  $\begin{array}{l}x=h+r\cos \left(\theta \right)\\ \Rightarrow x=2+\left(1\right)\cos \left(\theta \right)\\ \Rightarrow x=2+\cos \left(\theta \right)\\ y=k+r\sin \left(\theta \right)\\ \Rightarrow y=3+\left(1\right)\sin \left(\theta \right)\\ \Rightarrow y=3+\sin \left(\theta \right)\end{array}$

Therefore,

The parametric equation which represents given graph is

  $\begin{array}{l}x=2+\cos \left(\theta \right)\\ y=3+\sin \left(\theta \right)\end{array}$

Here circle equation will be

  $\begin{array}{l}\cos \left(\theta \right)=x-2\\ \sin \left(\theta \right)=y-3\end{array}$

Therefore,

Circle equation is

  ${\left(x-2\right)}^2+{\left(y-3\right)}^2=1$

This can also be written as

  ${\left(2-x\right)}^2+{\left(3-y\right)}^2=1$

Converting this to parametric

We get

  $\begin{array}{l}2-x=\cos \left(\theta \right)\\ 3-y=\sin \left(\theta \right)\end{array}$

  $\begin{array}{l}2-x=\cos \left(\theta \right)\\ \Rightarrow x=2-\cos \left(\theta \right)\\ 3-y=\sin \left(\theta \right)\\ \Rightarrow y=3-\sin \left(\theta \right)\end{array}$

Therefore,

Parametric equation of given graph is

  $\begin{array}{l}x=2-\cos \left(\theta \right)\\ y=3-\sin \left(\theta \right)\end{array}$