Determine the eccentricity, the type of conic, and the directrix.
-$$r = \frac { 5 } { 2 + 8 \sin \theta }$$
A) Eccentricity $= \frac { 8 } { 5 }$; ellipse; directrix $y = - 20$
B) Eccentricity $= 4$; ellipse; directrix $y = - \frac { 5 } { 4 }$
C) Eccentricity $= 4$; hyperbola; directrix $y = \frac { 5 } { 8 }$
D) Eccentricity $= \frac { 1 } { 4 } ;$ ellipse; directrix $\mathrm { y } = - 20$

Question

Quizplus · Accepted Answer

The given equation is in the form $ r = \frac{k}{1 + e \sin \theta} $, where $ e $ is the eccentricity. Here, $ e = 8/2 = 4 $, indicating a hyperbola since $ e > 1 $. The directrix is calculated as $ y = \frac{k}{e} = \frac{5}{8} $.

Determine the Eccentricity, the Type of Conic, and the Directrix r=52+8sin⁡θr = \frac { 5 } { 2 + 8 \sin \theta }r=2+8sinθ5​

Determine the Eccentricity, the Type of Conic, and the Directrix $r = \frac { 5 } { 2 + 8 \sin \theta }$