Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: $( \pm 4,0 )$, asymptotes: $y = \pm \frac { 7 } { 4 } x$
A) $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 49 } = 1$
B) $\frac { y ^ { 2 } } { 49 } - \frac { x ^ { 2 } } { 16 } = 1$
C) $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 49 } = 1$
D) $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 16 } = 1$

Question

Quizplus · Accepted Answer

The vertices at $(\pm 4,0)$ indicate that the hyperbola is centered at the origin and opens along the x-axis, which means the x-term will be positive in the equation. The distance from the center to a vertex is $a=4$, so $a^2=16$. The slope of the asymptotes $\pm \frac{7}{4}$ gives the relationship between $a$ and $b$ in $y = \pm \frac{b}{a}x$, leading to $b=7$ and $b^2=49$. Therefore, the correct equation is $\frac{x^2}{16} - \frac{y^2}{49} = 1$.

Find an Equation of the Hyperbola Centered at the Origin (±4,0)( \pm 4,0 )(±4,0)

Find an Equation of the Hyperbola Centered at the Origin $( \pm 4,0 )$