To find the equation that describes the hyperbola with a center at (0,0), a vertex of (0,19), and an asymptote of y = (19/16)x, we can use the standard form of the equation for a hyperbola:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
where (h, k) represents the center of the hyperbola, and a and b are positive real numbers representing the distances from the center to the vertices.
Given that the center is (0,0), we have h = 0 and k = 0.
The distance from the center to the vertex is given as 19, which corresponds to the value of b. Therefore, b = 19.
Now, let's find the value of a.
The equation of the asymptote, y = (19/16)x, can be written as:
y = (b/a)x
Comparing this with the standard form of the equation of the asymptote, y = mx, we have:
m = b/a
Substituting the values of b and m, we get:
(19/16) = 19/a
To solve for a, we can cross-multiply and solve the resulting equation:
19a = 16 * 19
19a = 304
a = 304/19
a = 16
Now that we have the values of a and b, we can substitute them into the standard form of the equation to obtain the equation that describes the hyperbola:
x^2 / (16^2) - y^2 / (19^2) = 1
Simplifying further, we have:
x^2 / 256 - y^2 / 361 = 1
Thus, the equation that describes the hyperbola is:
x^2 / 256 - y^2 / 361 = 1