Answer
$\displaystyle \frac{x^{2}}{9}-\frac{y^{2}}{16}=1$
Work Step by Step
Foci on the x-axis: $\quad \displaystyle \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
Center-to-focus distance: $\quad c=\sqrt{a^{2}+b^{2}}$
Foci: $\quad (\pm c, 0)$
Vertices: $\quad (\pm a, 0)$
Asymptotes: $\quad \displaystyle \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0 \quad$ or $\quad y=\displaystyle \pm\frac{b}{a}x$
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Given the vertices, $\quad a^{2}=9$
Given the asymptotes, $\quad \displaystyle \Rightarrow\quad(\frac{b}{a})^{2}=\frac{16}{9}\quad \Rightarrow\quad b^{2}=16$
The equation is $\displaystyle \frac{x^{2}}{9}-\frac{y^{2}}{16}=1$
