Find an equation for the hyperbola that has vertices $( \pm 3,0 )$ and asymptotes $y = \pm 5 x$
A) $\frac { x ^ { 2 } } { 225 } - \frac { y ^ { 2 } } { 9 } = 1$
B) $\frac { y ^ { 2 } } { 3 } - \frac { x ^ { 2 } } { 15 } = 1$
C) $x ^ { 2 } - \frac { y ^ { 2 } } { 5 } = 1$
D) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 225 } = 1$
E) none of these

Question

Quizplus · Accepted Answer

Since the center of the hyperbola is at the origin (since the asymptotes pass through the origin) and the distance from the center to each vertex is 3, the distance between each branch is 2(3) = 6. Therefore, the equation of the hyperbola must be of the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $a = 3$ and $b = \sqrt{3^2 - 5^2} = 4$. Plugging these values into the equation gives $\frac{x^2}{9} - \frac{y^2}{16} = 1$. Multiplying both sides by 144 gives $\frac{x^2}{9} - \frac{y^2}{225} = \boxed{D) \frac{x^2}{9} - \frac{y^2}{225} = 1}$.

Find an Equation for the Hyperbola That Has Vertices (±3,0)( \pm 3,0 )(±3,0)

Find an Equation for the Hyperbola That Has Vertices $( \pm 3,0 )$