A hyperbola centered at (0, 0) has vertices (15, 0) and (-15, 0) and one focus at (17, 0). What is the standard-form equation of the hyperbola? Find the correct equation in standard form and show correct, step-by-step work.
The standard form equation for a hyperbola centered at the origin is:
$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
where (a, 0) are the vertices and (c, 0) are the foci. The distance from the center to a vertex is a, and the distance from the center to a focus is c. The relationship between a, b, and c is given by the equation:
$c^2 = a^2 + b^2$
In this case, the center is at (0, 0), the vertices are at (15, 0) and (-15, 0), and one focus is at (17, 0).
Since the center is at the origin, a = 15 and c = 17. To find b, we use the relationship c^2 = a^2 + b^2:
$17^2 = 15^2 + b^2$
$289 = 225 + b^2$
$b^2 = 64$
$b = 8$
So, the equation of the hyperbola in standard form is:
$\frac{x^2}{225} - \frac{y^2}{64} = 1$