To find the length of intercept AB and the angle subtended by segment AB on the center of the circle, we need to first find the coordinates of points A and B.

Step 1: Finding the coordinates of points A and B
Given that line xy = 4 cuts the circle x² + y² - 6x - 8y - 11 = 0 at two points A and B, we can substitute the equation of the line into the equation of the circle to solve for the coordinates of A and B.

The equation of the line xy = 4 can be rewritten as y = 4/x.

Substituting this into the equation of the circle, we get:
x² + (4/x)² - 6x - 8(4/x) - 11 = 0

Simplifying the equation, we have:
x⁴ - 6x³ + 16x² - 88x - 176 = 0

We can solve this equation using numerical methods or a graphing calculator to find the values of x that satisfy the equation. Let's assume the values of x are x₁ and x₂.

Step 2: Calculating the length of intercept AB
Once we have the x-coordinates of A and B, we can substitute them into the equation of the line to find the corresponding y-coordinates. This will give us the coordinates of points A and B.

Let's say the coordinates of A are (x₁, y₁) and the coordinates of B are (x₂, y₂).

The length of intercept AB can be calculated using the distance formula:
AB = √[(x₂ - x₁)² + (y₂ - y₁)²]

Step 3: Calculating the angle subtended by segment AB on the center of the circle
To find the angle subtended by segment AB on the center of the circle, we can use the formula for the angle subtended by an arc of a circle:
θ = 2 * arcsin(AB/2r)

In this case, we need to find the radius of the circle. We can do this by completing the square for the equation of the circle.

The given equation of the circle is:
x² + y² - 6x - 8y - 11 = 0

Rearranging the terms, we get:
(x² - 6x + 9) + (y² - 8y + 16) - 36 - 16 - 11 = 0

(x - 3)² + (y - 4)² = 63

Comparing this equation with the standard equation of a circle, (x - h)² + (y - k)² = r², we can see that the center of the circle is at (3, 4) and the radius is √63.

Substituting the values of AB and r into the formula, we can calculate the angle subtended by segment AB on the center of the circle.

Note: The actual calculations for the coordinates of A and B, length AB, and the angle subtended may require numerical methods or a graphing calculator, as the equations involved are not easily solvable algebraically.

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