To determine the area of the triangle $\triangle OPQ$, we need to follow these steps:

1. Identify the coordinates of points $P$ and $Q$ on the graph of $f(x) = \frac{1}{x}$:

   • $P = (a, f(a)) = \left(a, \frac{1}{a}\right)$
   • $Q = \left(\frac{1}{a}, f\left(\frac{1}{a}\right)\right) = \left(\frac{1}{a}, a\right)$

2. Calculate the vertices of the triangle:

   • $O = (0, 0)$
   • $P = \left(a, \frac{1}{a}\right)$
   • $Q = \left(\frac{1}{a}, a\right)$

3. Use the determinant formula for the area of a triangle given its vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$: $\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$

4. Substitute the coordinates of $O$, $P$, and $Q$: $\text{Area} = \frac{1}{2} \left| 0 \left(\frac{1}{a} - a\right) + a \left(a - 0\right) + \frac{1}{a} \left(0 - \frac{1}{a}\right) \right|$

5. Simplify the expression: $\text{Area} = \frac{1}{2} \left| 0 + a^2 + \frac{1}{a} \left(-\frac{1}{a}\right) \right|$ $\text{Area} = \frac{1}{2} \left| a^2 - \frac{1}{a^2} \right|$

6. Final area calculation: $\text{Area} = \frac{1}{2} \left| a^2 - \frac{1}{a^2} \right|$

Answer:

The area of the triangle $\triangle OPQ$ in terms of $a$ is: $\boxed{\frac{1}{2} \left| a^2 - \frac{1}{a^2} \right|}$

If you have any further questions or need more details, feel free to ask! Here are some questions you might be interested in:

1. How can I verify the coordinates of points $P$ and $Q$?
2. Can you provide a detailed explanation of the determinant formula used for the area of a triangle?
3. What if the function $f(x)$ was different, how would that affect the area calculation?
4. Can you provide a graph for better visualization?
5. Are there alternative methods to solve this problem?

Tip: Always ensure the correct interpretation of the function and its points when solving geometry problems involving areas of triangles.