Select the answer that represents the hyperbola as well as its eccentricity, center and values of $a$ , $b$ and $c$ . $r = \frac { 5 } { 2 + 6 \cos \theta }$
A) Eccentricity: $3$ Center: $\left( \frac { 15 } { 16 } , 0 \right)$ $a = \frac { 5 } { 16 } , b = \frac { 5 \sqrt { 2 } } { 8 } , c = \frac { 15 } { 16 }$
B) Eccentricity: $3$ Center: $\left( - \frac { 15 } { 16 } , 0 \right)$ $a = \frac { 5 } { 16 } , b = \frac { 5 \sqrt { 2 } } { 8 } , c = \frac { 15 } { 16 }$
C) Eccentricity: $3$ Center: $\left( \frac { 1 } { 2 } , 0 \right)$ $a = \frac { 1 } { 6 } , b = \frac { \sqrt { 2 } } { 3 } , c = \frac { 1 } { 2 }$
D) Eccentricity: $3$ Center: $\left( - \frac { 1 } { 2 } , 0 \right)$ $a = \frac { 1 } { 6 } , b = \frac { \sqrt { 2 } } { 3 } , c = \frac { 1 } { 2 }$
E) Eccentricity: $3$ Center: $\left( \frac { 15 } { 16 } , 0 \right)$ $a = \frac { 1 } { 6 } , b = \frac { \sqrt { 2 } } { 3 } , c = \frac { 1 } { 2 }$

Question

Select the answer that represents the hyperbola as well as its eccentricity, center and values of $a$ , $b$ and $c$ . $r = \frac { 5 } { 2 + 6 \cos \theta }$
A) Eccentricity: $3$ Center: $\left( \frac { 15 } { 16 } , 0 \right)$ $a = \frac { 5 } { 16 } , b = \frac { 5 \sqrt { 2 } } { 8 } , c = \frac { 15 } { 16 }$ 
B) Eccentricity: $3$ Center: $\left( - \frac { 15 } { 16 } , 0 \right)$ $a = \frac { 5 } { 16 } , b = \frac { 5 \sqrt { 2 } } { 8 } , c = \frac { 15 } { 16 }$ 
C) Eccentricity: $3$ Center: $\left( \frac { 1 } { 2 } , 0 \right)$ $a = \frac { 1 } { 6 } , b = \frac { \sqrt { 2 } } { 3 } , c = \frac { 1 } { 2 }$ 
D) Eccentricity: $3$ Center: $\left( - \frac { 1 } { 2 } , 0 \right)$ $a = \frac { 1 } { 6 } , b = \frac { \sqrt { 2 } } { 3 } , c = \frac { 1 } { 2 }$ 
E) Eccentricity: $3$ Center: $\left( \frac { 15 } { 16 } , 0 \right)$ $a = \frac { 1 } { 6 } , b = \frac { \sqrt { 2 } } { 3 } , c = \frac { 1 } { 2 }$

Quizplus · Accepted Answer

The given polar equation represents a hyperbola with center at $(h,k)=(\frac{15}{16},0)$, and the eccentricity is given by $e=\sqrt{1+\frac{b^2}{a^2}}=\sqrt{1+\frac{2}{25}}=3/5$. The standard form of a hyperbola centered at $(h,k)$ is
$$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1,$$
where $a>c$ and $c$ is the distance from the center to each of the vertices. Comparing this form with the given polar equation,
$$\frac{(x-\frac{15}{16})^2}{(\frac{5}{16})^2}-\frac{y^2}{(\frac{5\sqrt{2}}{8})^2}=1,$$
we find that $a=\frac{5}{16}$, $b=\frac{5\sqrt{2}}{8}$, and $c=\frac{15}{16}$. Therefore, the correct choice is $\boxed{\textbf{(A)}}$.

Select the Answer That Represents the Hyperbola as Well as Its