Final answer:
To find the arc length of the hyperbola xy = 1, we set up an integral with respect to x, after parameterizing the curve with x and solving for y as y = 1/x. Using the formula for arc length and the derivative of y, we obtain an integral to calculate the length between points (1, 4) and (2, 2) as these satisfy the equation of the hyperbola.
Step-by-step explanation:
To find the arc length of a hyperbola xy = 1, you need to set up an integral that measures the length of the curve between two points. The form of the integral depends on the parameterization of the hyperbola. In this case, we can parameterize using x as the parameter and solve for y as y = 1/x.
The formula for the arc length L of a curve from one point to another, defined as y = f(x), in the interval [a, b] is given by:
L = ∫ab √(1 + (f'(x))2) dx
For our hyperbola xy = 1, we have y = 1/x, and the derivative y' = -1/x2. Substituting into the arc length formula gives:
L = ∫ab √(1 + (-1/x2)2) dx = ∫ab √(1 + 1/x4) dx
The correct end points must satisfy the hyperbola's equation, so Options 1 and 3 can be disregarded because (-1, -4) and (3, 9) don't satisfy xy = 1. The points (1, 4) and (2, 2) do satisfy the equation, so we would set up our integral with limits from 1 to 2:
L = ∫12 √(1 + 1/x4) dx