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The question asks to find the value of \(a - b\), where points \(P\) and \(Q\) divide the line segment \(AB\) into three equal parts.

### Given:
- Coordinates of \(A = (0, 5)\),
- Coordinates of \(B = (-3, -1)\),
- Points \(P\) and \(Q\) divide \(AB\) into three equal parts.

Let’s calculate the coordinates of points \(P\) and \(Q\) using the section formula. 

#### Step 1: Find the coordinates of \(P\) and \(Q\)

Since \(P\) and \(Q\) divide \(AB\) into three equal parts, they divide \(AB\) in the ratio \(1:2\) and \(2:1\) respectively.

1. **For Point \(P\):** (divides \(AB\) in the ratio 1:2)
   Using the section formula, the coordinates of \(P\) are:
   \[
   P = \left( \frac{2(0) + 1(-3)}{1+2}, \frac{2(5) + 1(-1)}{1+2} \right)
   \]
   Simplifying:
   \[
   P = \left( \frac{-3}{3}, \frac{10 - 1}{3} \right) = (-1, 3)
   \]

2. **For Point \(Q\):** (divides \(AB\) in the ratio 2:1)
   Using the section formula, the coordinates of \(Q\) are:
   \[
   Q = \left( \frac{1(0) + 2(-3)}{1+2}, \frac{1(5) + 2(-1)}{1+2} \right)
   \]
   Simplifying:
   \[
   Q = \left( \frac{-6}{3}, \frac{5 - 2}{3} \right) = (-2, 1)
   \]

#### Step 2: Interpretation of \(a\) and \(b\)

From the diagram:
- The coordinates of point \(P\) are \( (-1, a) \), so \(a = 3\).
- The coordinates of point \(Q\) are \( (-2, b) \), so \(b = 1\).

#### Step 3: Find \(a - b\)
\[
a - b = 3 - 1 = 2
\]

### Final Answer:
The value of \(a - b\) is \( \boxed{2} \).

Would you like further details or have any questions? Here are a few related questions:

1. How is the section formula derived in coordinate geometry?
2. What is the geometric interpretation of dividing a line segment in a given ratio?
3. Can the section formula be extended to 3D space? How?
4. How do you use vectors to find points dividing a segment into equal parts?
5. What are other real-life applications of the section formula?

**Tip:** Always verify that the points calculated align with the given diagram or coordinates to avoid errors in interpretation.

In the following figure, points P and Q divide line segment AB into three equal parts. The value of a - b is?

Solve the coordinate geometry problem where points P and Q divide the line segment AB into three equal parts. Learn how to use the section formula to find the coordinates of P and Q, and calculate the value of a - b. Ideal for students in Grades 9-11.

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