Write the standard form of the equation of the hyperbola centered at the origin. Vertices: $( - 2,0 ) , ( 2,0 )$ Asymptotes: $y = 5 x , y = - 5 x$
A) $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 4 / 25 } = 1$
B) $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 100 } = 1$
C) $\frac { x ^ { 2 } } { 4 / 25 } - \frac { y ^ { 2 } } { 4 } = 1$
D) $\frac { y ^ { 2 } } { 100 } - \frac { x ^ { 2 } } { 4 } = 1$
E) $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 4 } = 1$

Question

Quizplus · Accepted Answer

The standard form of a hyperbola centered at the origin with a horizontal transverse axis is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $2a$ is the distance between the vertices, and $b^2$ can be found from the equation of the asymptotes $y = \pm \frac{b}{a}x$. Given vertices at $(-2,0)$ and $(2,0)$, $a = 2$, so $a^2 = 4$. The asymptotes are given as $y = \pm 5x$, implying $\frac{b}{a} = 5$, so $b = 5a = 10$, and $b^2 = 100$. Thus, the correct equation is $\frac{x^2}{4} - \frac{y^2}{100} = 1$.

Write the Standard Form of the Equation of the Hyperbola (−2,0),(2,0)( - 2,0 ) , ( 2,0 )(−2,0),(2,0)

Write the Standard Form of the Equation of the Hyperbola $( - 2,0 ) , ( 2,0 )$