Find the center, foci, and asymptotes of the hyperbola.
-$$\frac { y ^ { 2 } } { 256 } - \frac { x ^ { 2 } } { 144 } = 1$$
A) C: $( 0,0 ) ; F : ( 0 , - 20 ) , ( 0,20 ) ; A : y = \frac { 4 } { 3 } x , y = - \frac { 4 } { 3 } x$
B) C: $( 0,0 ) ; \mathrm { F } : ( - 12,0 ) , ( 12,0 ) ; \mathrm { A } : \mathrm { y } = \frac { 3 } { 4 } \mathrm { x } , \mathrm { y } = - \frac { 3 } { 4 } \mathrm { x }$
C) C: $( 0,0 ) ; F : ( - 20,0 ) , ( 20,0 ) ; A : y = \frac { 3 } { 4 } x , y = - \frac { 3 } { 4 } x$
D) C: $( 0,0 ) ; F : ( 0 , - 12 ) , ( 0,12 ) ; A : y = \frac { 4 } { 3 } x , y = - \frac { 4 } { 3 } x$

Question

Quizplus · Accepted Answer

The center of the hyperbola is at the origin (0,0) since there are no shifts in the equation. The foci are found using $c^2 = a^2 + b^2$, where $a^2 = 256$ and $b^2 = 144$, giving $c = \sqrt{256 + 144} = 20$. Since $y^2$ is in the numerator, the foci are along the y-axis at (0, ±20). The asymptotes for a hyperbola of the form $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ are $y = \pm\frac{a}{b}x$, giving $y = \pm\frac{16}{12}x = \pm\frac{4}{3}x$.

Find the Center, Foci, and Asymptotes of the Hyperbola y2256−x2144=1\frac { y ^ { 2 } } { 256 } - \frac { x ^ { 2 } } { 144 } = 1256y2​−144x2​=1

Find the Center, Foci, and Asymptotes of the Hyperbola $\frac { y ^ { 2 } } { 256 } - \frac { x ^ { 2 } } { 144 } = 1$