Final answer:
The equation for a hyperbola with the center at the origin, vertex at (4,0), and focus at (6,0) is ((x^2)/(16)) - ((y^2)/(20)) = 1. By finding the values of a and c from the given points, we then use the relationship c^2 = a^2 + b^2 to solve for b and determine the standard form equation.
Step-by-step explanation:
To write the equation of a hyperbola in standard form, we need to know two key pieces of information about its structure: the distances to the vertices from the center, known as a, and the distances to the foci from the center, known as c. Given that the center of the hyperbola is at (0,0), the vertex at (4,0), and the focus at (6,0), we can infer that a = 4 and c = 6. Notice that the hyperbola is centered at the origin, which simplifies the algebra.
The standard form equation for a hyperbola with a horizontal transverse axis is given as ((x^2)/(a^2)) - ((y^2)/(b^2)) = 1. Here, b can be found using the relationship c^2 = a^2 + b^2. Substituting a = 4 and c = 6 into this equation, we get 6^2 = 4^2 + b^2, which means b^2 = 36 - 16 = 20.
Therefore, the standard form of the hyperbola is ((x^2)/(16)) - ((y^2)/(20)) = 1. This equation describes a hyperbola centered at the origin, with vertices along the x-axis and opens to the left and right due to the positive x term in the equation.