The curve C has equation y = 9 - 4x - \frac{8}{x}, \, x > 0 - Edexcel - A-Level Maths Pure - Question 5 - 2009 - Paper 1

To calculate the area of triangle APB, we need the coordinates of points A and B:

1. Finding point A: Setting y = 0 in the tangent equation: \Rightarrow 2x_A = 1\ \Rightarrow x_A = \frac{1}{2}.$$ Thus, A is at (0.5, 0).
2. Finding point B: Setting y = 0 in the normal equation: \Rightarrow x_B = 8.$$ Thus, B is at (8, 0).

Now we can calculate the area of triangle APB using the formula:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

The base AB is: $AB = x_B - x_A = 8 - 0.5 = 7.5.$
The height is the y-coordinate of P, which is -3. Therefore,

$\text{Area} = \frac{1}{2} \times 7.5 \times 3 = \frac{22.5}{2} = 11.25.$

The area of triangle APB is 11.25.