Write an equation for the hyperbola.
-vertices $( - 6 , - 2 ) , ( - 6,16 )$; eccentricity $\frac { \sqrt { 106 } } { 9 }$
A) $\frac { ( y - 6 ) ^ { 2 } } { 81 } - \frac { ( x + 7 ) ^ { 2 } } { 25 } = 1$
B) $\frac { ( y - 7 ) ^ { 2 } } { 81 } - \frac { ( x + 6 ) ^ { 2 } } { 25 } = 1$
C) $\frac { ( y - 6 ) ^ { 2 } } { 81 } - \frac { ( x - 7 ) ^ { 2 } } { 25 } = 1$
D) $\frac { ( y + 6 ) ^ { 2 } } { 25 } - \frac { ( x - 7 ) ^ { 2 } } { 81 } = 1$

Question

Quizplus · Accepted Answer

The vertices indicate that the hyperbola is vertical, centered at $(-6,7)$ with $2a=18$, so $a=9$ and $a^2=81$. The eccentricity formula $e=c/a$ with $e=\frac{\sqrt{106}}{9}$ implies $c^2=106$. Since $c^2=a^2+b^2$, we find $b^2=106-81=25$. Thus, the correct equation is $\frac{(y-7)^2}{81}-\frac{(x+6)^2}{25}=1$.

Write an Equation for the Hyperbola (−6,−2),(−6,16)( - 6 , - 2 ) , ( - 6,16 )(−6,−2),(−6,16)

Write an Equation for the Hyperbola $( - 6 , - 2 ) , ( - 6,16 )$