Let P and Q be points on the line joining A(-2, 5) and B(3, 1) such that AP = PQ = QB. If mid-point of PQ is (a, b), then the value of `(b)/(a)` is

Answer

To solve the problem, we need to find the coordinates of points P and Q that trisect the line segment AB, where A(-2, 5) and B(3, 1). Then we will calculate the midpoint of PQ and find the value of \( \frac{b}{a} \).

### Step 1: Find the coordinates of points A and B
We have:
- A = (-2, 5)
- B = (3, 1)

### Step 2: Calculate the distance between points A and B
The distance \( d \) between points A and B can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of A and B:
\[ d = \sqrt{(3 - (-2))^2 + (1 - 5)^2} = \sqrt{(3 + 2)^2 + (1 - 5)^2} = \sqrt{5^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41} \]

### Step 3: Find the coordinates of points P and Q
Since AP = PQ = QB, the points P and Q trisect the line segment AB. This means that the segment AB is divided into three equal parts.

The coordinates of point P can be found using the section formula:
\[ P = \left( \frac{2x_1 + x_2}{3}, \frac{2y_1 + y_2}{3} \right) \]
Substituting the coordinates of A and B:
\[ P = \left( \frac{2(-2) + 3}{3}, \frac{2(5) + 1}{3} \right) = \left( \frac{-4 + 3}{3}, \frac{10 + 1}{3} \right) = \left( \frac{-1}{3}, \frac{11}{3} \right) \]

Similarly, the coordinates of point Q can be found as:
\[ Q = \left( \frac{x_1 + 2x_2}{3}, \frac{y_1 + 2y_2}{3} \right) \]
Substituting the coordinates of A and B:
\[ Q = \left( \frac{-2 + 2(3)}{3}, \frac{5 + 2(1)}{3} \right) = \left( \frac{-2 + 6}{3}, \frac{5 + 2}{3} \right) = \left( \frac{4}{3}, \frac{7}{3} \right) \]

### Step 4: Find the midpoint of PQ
The midpoint M of PQ can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of P and Q:
\[ M = \left( \frac{\frac{-1}{3} + \frac{4}{3}}{2}, \frac{\frac{11}{3} + \frac{7}{3}}{2} \right) = \left( \frac{\frac{3}{3}}{2}, \frac{\frac{18}{3}}{2} \right) = \left( \frac{1}{2}, 3 \right) \]

### Step 5: Calculate \( \frac{b}{a} \)
From the midpoint M, we have:
- \( a = \frac{1}{2} \)
- \( b = 3 \)

Now, we can calculate \( \frac{b}{a} \):
\[ \frac{b}{a} = \frac{3}{\frac{1}{2}} = 3 \times 2 = 6 \]

### Final Answer
The value of \( \frac{b}{a} \) is \( 6 \).