Mathematics

A hyperbola is defined by the relationship between its foci and the distance across its center along the transverse axis. Here, you know the foci locations tell you $c = 5$, and the transverse axis length gives you $2a = 8$, so $a = 4$. The key insight is using the fundamental hyperbola identity $c^2 = a^2 + b^2$ to find $b^2$, then plugging into the standard form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (since the foci lie on the $x$-axis).

A hyperbola is defined by the relationship between its foci and the distance across its center along the transverse axis. Here, you know the foci locations tell you $c = 5$, and the transverse axis length gives you $2a = 8$, so $a = 4$. The key insight is using the fundamental hyperbola identity $c^2 = a^2 + b^2$ to find $b^2$, then plugging into the standard form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (since the foci lie on the $x$-axis).