Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin.
-$r = \frac { 24 } { 8 - 4 \sin \theta }$

Question

Quizplus · Accepted Answer

The given equation is in polar form, which represents an ellipse with foci at the origin and at a distance of $\frac{c}{a} = 2$ from the origin along the major axis. To graph the ellipse, we can convert the polar equation into Cartesian coordinates:

\begin{align*}
r &= \frac { 24 } { 8 - 4 \sin 	heta }\
r(8 - 4\sin	heta) &= 24 \
8r - 4r\sin	heta &= 24\
r &= \frac{3}{2}\sin	heta + 2.
\end{align*}

Using the conversion $x = r\cos	heta$, $y = r\sin	heta$, we get:

$$x = \left(\frac{3}{2}\sin	heta + 2ight)\cos	heta,\quad y = \left(\frac{3}{2}\sin	heta + 2ight)\sin	heta.$$

Simplifying these expressions we get:

$$\frac{(x-4)^2}{\frac{27}{4}} + \frac{y^2}{\frac{27}{4}} = 1,$$

which is the equation of an ellipse centered at $(4,0)$ with semi-axes of length $\frac{3}{2}$ and $\frac{3\sqrt{3}}{2}$.

The directrix for this ellipse is the line $x = -\frac{27}{4}$, which is the line that is at a distance of $\frac{2a}{c} = \frac{4}{2} = 2$ from the center of the ellipse along the major axis.

Therefore, choice (B) is the best answer.

Graph the Parabola or Ellipse r=248−4sin⁡θr = \frac { 24 } { 8 - 4 \sin \theta }r=8−4sinθ24​

Graph the Parabola or Ellipse $r = \frac { 24 } { 8 - 4 \sin \theta }$