Chapter 9, Problem 10RE

(a)

To determine

To find:The equationin rectangular coordinates for given polar coordinate.

Answer to Problem 10RE

The equation in rectangular coordinates is $x^2+y^2-8x+4y-5=0$ .

Explanation of Solution

Given information:

The given polar equation is $r^2+4r\sin \theta -8r\cos \theta =5$ .

Calculation:

The relation between polar coordinates $\left(r,\theta \right)$ , and rectangular coordinates is.

  $\begin{array}{c}{r}^2=x^2+y^2\\ \tan \theta =\frac{y}{x}\end{array}$

The sine function is.

  $\sin \theta =\frac{y}{\sqrt{x^2+y^2}}$

The cosine function is.

  $\cos \theta =\frac{x}{\sqrt{x^2+y^2}}$

Substitute the values in given equation.

  $x^2+y^2+4\left(\sqrt{x^2+y^2}\right)\left(\frac{y}{\sqrt{x^2+y^2}}\right)-8\left(\sqrt{x^2+y^2}\right)\left(\frac{x}{\sqrt{x^2+y^2}}\right)=5$

Solve the above equation

  $x^2+y^2-8x+4y-5=0$

Therefore, the equation in rectangular coordinates is $x^2+y^2-8x+4y-5=0$ .

(b)

To determine

To graph:For the equation.

Explanation of Solution

Given information:

The equation is $x^2+y^2-8x+4y-5=0$ .

Graph:

This equation $x^2+y^2-8x+4y-5=0$ is the equation of circle with center $\left(4,2\right)$ .

The graph for the equation is shown in figure (1).

  

Figure (1)

Interpretation:Graph for the equation $x^2+y^2-8x+4y-5=0$ is shown in figure (1).