Identify the foci and write equations for the asymptotes.
-$$9 ( y - 4 ) ^ { 2 } - 100 ( x + 4 ) ^ { 2 } = - 900$$
A) foci: $( - 4 + \sqrt { 109 } , 4 ) , ( - 4 - \sqrt { 109 } , 4 )$
asymptotes: $y = \frac { 10 } { 3 } x + \frac { 52 } { 3 }$ and $y = \frac { 10 } { 3 } x - \frac { 28 } { 3 }$
B) foci: $( 0 , - 4 + \sqrt { 109 } ) , ( 0 , - 4 - \sqrt { 109 } )$
asymptotes: $y = \frac { 3 } { 10 } x - \frac { 14 } { 5 }$ and $y = - \frac { 3 } { 10 } x - \frac { 26 } { 5 }$
C) foci: $( 4 + \sqrt { 109 } , 0 ) , ( 4 - \sqrt { 109 } , 0 )$
asymptotes: $y = \frac { 10 } { 3 } x - \frac { 28 } { 3 }$ and $y = \frac { 10 } { 3 } x + \frac { 52 } { 3 }$
D) foci: $( 0,4 + \sqrt { 109 } ) , ( 0,4 - \sqrt { 109 } )$
asymptotes: $y = \frac { 3 } { 10 } x + \frac { 26 } { 5 }$ and $y = - \frac { 3 } { 10 } x + \frac { 14 } { 5 }$

Question

Identify the foci and write equations for the asymptotes.
-$$9 ( y - 4 ) ^ { 2 } - 100 ( x + 4 ) ^ { 2 } = - 900$$
A) foci: $( - 4 + \sqrt { 109 } , 4 ) , ( - 4 - \sqrt { 109 } , 4 )$
 asymptotes: $y = \frac { 10 } { 3 } x + \frac { 52 } { 3 }$ and $y = \frac { 10 } { 3 } x - \frac { 28 } { 3 }$
B) foci: $( 0 , - 4 + \sqrt { 109 } ) , ( 0 , - 4 - \sqrt { 109 } )$
 asymptotes: $y = \frac { 3 } { 10 } x - \frac { 14 } { 5 }$ and $y = - \frac { 3 } { 10 } x - \frac { 26 } { 5 }$
C) foci: $( 4 + \sqrt { 109 } , 0 ) , ( 4 - \sqrt { 109 } , 0 )$
 asymptotes: $y = \frac { 10 } { 3 } x - \frac { 28 } { 3 }$ and $y = \frac { 10 } { 3 } x + \frac { 52 } { 3 }$
D) foci: $( 0,4 + \sqrt { 109 } ) , ( 0,4 - \sqrt { 109 } )$
 asymptotes: $y = \frac { 3 } { 10 } x + \frac { 26 } { 5 }$ and $y = - \frac { 3 } { 10 } x + \frac { 14 } { 5 }$

Quizplus · Accepted Answer

The equation is in the form $(y - k)^2/b^2 - (x - h)^2/a^2 = 1$ with $a^2 = 100$ and $b^2 = 9$. So, $a = 10$ and $b = 3$. The foci are $(h \pm c, k)$ where $c^2 = a^2 + b^2$. So, $c^2 = 100 + 9 = 109$ and $c = \sqrt{109}$. Therefore, the foci are $(-4 \pm \sqrt{109}, 4)$. The asymptotes are $y = k \pm (a/b)(x - h)$. So, the asymptotes are $y = 4 \pm (10/3)(x + 4)$, which simplifies to $y = (10/3)x + (52/3)$ and $y = (10/3)x - (28/3)$.

Identify the Foci and Write Equations for the Asymptotes 9(y−4)2−100(x+4)2=−9009 ( y - 4 ) ^ { 2 } - 100 ( x + 4 ) ^ { 2 } = - 9009(y−4)2−100(x+4)2=−900

Identify the Foci and Write Equations for the Asymptotes $9 ( y - 4 ) ^ { 2 } - 100 ( x + 4 ) ^ { 2 } = - 900$