Chapter 10.3, Problem 25E

To determine

To replace: the polar equation $r=\frac{5}{\sin \theta -2\cos \theta }$ by an equivalent Cartesian(rectangular) equation and identify or describe the graph without using a grapher.

Answer to Problem 25E

The cartesian equation equivalent to the polar equation $r=\frac{5}{\sin \theta -2\cos \theta }$ is $y=2x+5$ .

It is an equation of a straight line with slope $2$ and $y$ -intercept is $5$ .

Explanation of Solution

Given information:

The polar equation $r=\frac{5}{\sin \theta -2\cos \theta }$ .

Formula used:

  $x=r\cos \theta$ and $y=r\sin \theta$ .

The formula to convert polar equation to cartesian equation: $r^2=x^2+y^2$ .

Calculation:

The given equation is $r=\frac{5}{\sin \theta -2\cos \theta }$

Multiply both sides of the equation by $\sin \theta -2\cos \theta$ .

  $\begin{array}{l}r\left(\sin \theta -2\cos \theta \right)=\frac{5}{\sin \theta -2\cos \theta }\left(\sin \theta -2\cos \theta \right)\\ r\left(\sin \theta -2\cos \theta \right)=5\\ r\sin \theta -2r\cos \theta =5\end{array}$

Substitute $x$ for $r\cos \theta$ and $y$ for $r\sin \theta$ in $r\sin \theta -2r\cos \theta =5$ .

  $y-2x=5$

Write the equation in slope-intercept form.

  $y=2x+5$

So, it is a straight line with slope $2$ and $y$ -intercept is $5$ .

The cartesian equation equivalent to the polar equation $r=\frac{5}{\sin \theta -2\cos \theta }$ is $y=2x+5$ .

It is an equation of a straight line with slope $2$ and $y$ -intercept is $5$ .