Question

Identify the conic that the polar equation represents. Also, give the position of the directrix.
$r=\frac{9}{1-3\cos \theta }$

• A. hyperbola, directrix perpendicular to the polar axis $3$ left of the pole
• B. ellipse, directrix perpendicular to the polar axis $3$ left of the pole
• C. ellipse, directrix perpendicular to the polar axis $3$ right of the pole
• D. hyperbola, directrix perpendicular to the polar axis $3$ right of the pole

Answer 1

The correct option is A hyperbola, directrix perpendicular to the polar axis $3$ left of the pole

Distance from $F$ to directrix $=2p=k$

$e=\frac{PF}{PD}\Rightarrow ePD=PFr[k-r\cos \theta ]=rek-er\cos \theta =rek=r+er\cos \theta \frac{ek}{1+e\cos \theta }=\frac{r(1+e\cos \theta )}{(1+e\cos \theta )}\therefore r=\frac{ek}{(1+e\cos \theta )}$

this is a polar equation.

$r=\frac{ke}{1\pm e\cos \theta }$ shows the vertical directrix where $e>0$ and $k=$ distance between focus(pole) and directrix

$\therefore r=\frac{9}{1-3\cos \theta }$

from this, $e=3$ and $k=3$

Thus, it is a hyperbola, directrix perpendicular to the polar axis $3$ left to the pole.