Find the center, foci, and asymptotes of the hyperbola.
-$\frac { x ^ { 2 } } { 225 } - \frac { y ^ { 2 } } { 400 } = 1$
A) C: $( 0,0 ) ; \mathrm { F } : ( - 25,0 ) , ( 25,0 ) ; \mathrm { A } : \mathrm { y } = \frac { 4 } { 3 } \mathrm { x } , \mathrm { y } = - \frac { 4 } { 3 } \mathrm { x }$
B) C: $( 0,0 ) ; \mathrm { F } : ( 0 , - 20 ) , ( 0,20 ) ; \mathrm { A } : \mathrm { y } = \frac { 3 } { 4 } \mathrm { x } , \mathrm { y } = - \frac { 3 } { 4 } \mathrm { x }$
C) C: $( 0,0 ) ; \mathrm { F } : ( - 25,0 ) , ( 25,0 ) ; \mathrm { A } : \mathrm { y } = \frac { 3 } { 4 } \mathrm { x } , \mathrm { y } = - \frac { 3 } { 4 } \mathrm { x }$
D) C: $( 0,0 ) ; F : ( 0 , - 20 ) , ( 0,20 ) ; A : y = \frac { 4 } { 3 } x , y = - x$

Question

Quizplus · Accepted Answer

The center of the hyperbola is at the origin (0,0) because there are no shifts in the equation. The foci are found using $c^2 = a^2 + b^2$, where $a^2 = 225$ and $b^2 = 400$, leading to $c = \sqrt{225 + 400} = 25$, so the foci are at $(-25,0)$ and $(25,0)$ because the x-term is positive, indicating a horizontal orientation. The asymptotes for a hyperbola of the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ are $y = \pm \frac{b}{a}x$, so here they are $y = \pm \frac{4}{3}x$.

Find the Center, Foci, and Asymptotes of the Hyperbola x2225−y2400=1\frac { x ^ { 2 } } { 225 } - \frac { y ^ { 2 } } { 400 } = 1225x2​−400y2​=1

Find the Center, Foci, and Asymptotes of the Hyperbola $\frac { x ^ { 2 } } { 225 } - \frac { y ^ { 2 } } { 400 } = 1$