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.