Determine the eccentricity, the type of conic, and the directrix.
-$$r = \frac { 7 } { 9 - 6 \cos \theta }$$
A) Eccentricity $= \frac { 2 } { 3 }$; ellipse; directrix $x = - \frac { 7 } { 6 }$
B) Eccentricity $= \frac { 2 } { 3 } ;$ hyperbola; directrix $x = - \frac { 21 } { 2 }$
C) Eccentricity $= \frac { 3 } { 2 }$; ellipse; directrix $x = - \frac { 7 } { 6 }$
D) Eccentricity $= \frac { 6 } { 7 }$; hyperbola; directrix $y = - \frac { 14 } { 3 }$

Question

Quizplus · Accepted Answer

The given equation is in the polar form $r = \frac{e d}{1 - e \cos \theta}$, where $e$ is the eccentricity and $d$ is the distance to the directrix. Comparing, $e = \frac{6}{9} = \frac{2}{3}$ and $d = 7$, indicating an ellipse (since $e < 1$). The directrix, for an ellipse in polar coordinates, is given by $x = -\frac{d}{e}$, which simplifies to $x = -\frac{7}{6}$.

Determine the Eccentricity, the Type of Conic, and the Directrix r=79−6cos⁡θr = \frac { 7 } { 9 - 6 \cos \theta }r=9−6cosθ7​

Determine the Eccentricity, the Type of Conic, and the Directrix $r = \frac { 7 } { 9 - 6 \cos \theta }$