A hyperbola is defined by the equation (y-4)^2/25 - (x-8)^2 = 1
To understand the properties of the given hyperbola defined by the equation:
1. Start by determining the center of the hyperbola. The equation is in the form (y-k)^2/a^2 - (x-h)^2/b^2 = 1, where (h, k) represents the center. In this case, the center is at (8, 4).
2. The distance between the center and the vertices can be found using the equation c^2 = a^2 + b^2, where c represents the distance from the center to the foci. In this case, a = 5 (since a^2 = 25), b = 1 (since b^2 = 1). Plugging these values, we have c^2 = 25 + 1 = 26. Thus, c = sqrt(26).
3. The vertices of the hyperbola can be calculated by adding and subtracting the value of "a" from the x-coordinate of the center. The vertices would be located at (8 - 5, 4) = (3, 4) and (8 + 5, 4) = (13, 4).
4. The foci of the hyperbola can be found by adding and subtracting the value of "c" from the x-coordinate of the center. The foci would be at (8 - sqrt(26), 4) and (8 + sqrt(26), 4), which can be approximated as (3.10, 4) and (12.90, 4).
5. To sketch the hyperbola, plot the center, vertices, and foci on a graph, then draw the curve defined by the equation. The hyperbola opens horizontally, meaning it extends infinitely along the y-axis, while the x-values approach (8 - sqrt(26)) and (8 + sqrt(26)).
These steps should help you understand the properties and graph of the given hyperbola.