Final answer:
The standard form of the equation of the hyperbola with the given vertices and focus is (x-4)²/16 - (y-1)²/33 = 1, calculated by identifying the center, and the lengths of the semi-transverse and semi-conjugate axes using distances between the vertices and the focus.
Step-by-step explanation:
To find the standard form of the equation of a hyperbola with vertices at (0,1) and (8,1) and one focus at (-3,1), we need to identify the center, the lengths of the transverse and conjugate axes, and the distance to the foci.
The vertices being on the same horizontal line and equidistant from the center suggest that the center is at the midpoint of the vertices which is at (4, 1). Since one vertex is at (0, 1) and the other at (8, 1), the distance between the vertices is 8 units. This distance is 2a, where a is the length of the semi-transverse axis, so we have a = 4. The focus (-3, 1) is 7 units away from the center, which means c, the distance from the center to a focus, is 7.
We need to find b, the length of the semi-conjugate axis. Using the relationship c² = a² + b², we plug in the known values and solve for b:
c² = a² + b²
7² = 4² + b²
49 = 16 + b²
33 = b²
b = √33
The standard form of the equation for a horizontal hyperbola (since the vertices are along a horizontal line) is:
∛((x-h)²/a²) - ∛((y-k)²/b²) = 1
where (h, k) is the center.
Thus, the standard form for our hyperbola is:
∛((x-4)²/4²) - ∛((y-1)²/√33²) = 1
or
(x-4)²/16 - (y-1)²/33 = 1