The graph of \[r = \frac{7}{3 \cos \theta + 2 \sin \theta}\] is a line. Find the slope of this line.

r = √ [ x^2 + y^2 ]

cos theta =   x/r =  x / √ [ x^2 + y^2 ]

sin theta = y/r = y / √ [ x^2 + y^2 ]

So we have

√ [ x^2 + y^2 ]   =                 7

                                  _____________

                                    [  3x +  2y ]

                                   ___________

                                   √ [ x^2 + y^2 ]

√ [ x^2 + y^2 ]  =     7 √ [ x^2 + y^2 ]

                              _____________

                                    3x   + 2y

1     =          7

              _______

               3x + 2y

3x + 2y  =   7

In the form Ax + By = C.....the slope is     -A / B

So.....slope   =        -3/2

The graph of
\(\huge{r = \dfrac{7}{3 \cos \theta + 2 \sin \theta}}\)
is a line. Find the slope of this line.

\(\begin{array}{|rcll|} \hline \mathbf{r(\theta)} &\mathbf{=}& \mathbf{\dfrac{7}{3 \cos \theta + 2 \sin \theta}} \\ \hline r(0) &=& \dfrac{7}{3 \cos(0) + 2 \sin(0)} \\\\ r(0) &=& \dfrac{7}{3} \\ \hline r(90) &=& \dfrac{7}{3 \cos(90) + 2 \sin(90)} \\\\ r(90) &=& \dfrac{7}{2} \\ \hline -m &=& \dfrac{r(90)}{r(0)} \\\\ &=& \dfrac{\dfrac{7}{2}}{ \dfrac{7}{3}} \\\\ -m &=& \dfrac{3}{2} \\\\ \mathbf{m} & \mathbf{=} & \mathbf{-\dfrac{3}{2}} \\ \hline \end{array}\)