Chapter 8, Problem 13RE

(a)

To determine

The given equation needs to write down in the form of polar-co-ordinates.

  $x+y=4$

Answer to Problem 13RE

The equation of the rectangular co-ordinates converts into a polar co-ordinates as $\cos \theta +\sin \theta =\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$r$}\right.$ .

Explanation of Solution

Given:

  $x+y=4$

Concept Used:

When a point is given with its rectangular co-ordinates $\left(x,y\right)$ , its polar co-ordinates $\left(r,\theta \right)$ are defined as:

  $\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array}$

Calculation:

The given equation is:

  $x+y=4$

Now need to convert the equation of rectangular co-ordinates into polar co-ordinates as:

  $\begin{array}{l}x+y=4\\ r\cos \theta +r\sin \theta =4\\ r\left(\cos \theta +\sin \theta \right)=4\\ r=\frac{4}{\cos \theta +\sin \theta }\end{array}$

Conclusion:

Hence, the simplified equation is $r=\frac{4}{\cos \theta +\sin \theta }$ .

(b)

To determine

The graph of the equation is need to drawn.

Explanation of Solution

Given:

  $x+y=4$

Graph:

  

Interpretation:

From the given data point:

The given equation is in the rectangular co-ordinates is drawn as above:

  $x+y=4$