In what ratio does the point `M(p-1)` divide the line segment joining the points `A(1,-3) and B(6,2)` ? Hence, find the value of p.

Answer

To solve the problem, we need to determine the ratio in which the point \( M(p, -1) \) divides the line segment joining the points \( A(1, -3) \) and \( B(6, 2) \). We will also find the value of \( p \).

### Step-by-Step Solution:

1. **Identify the coordinates of points A and B:**
- \( A(1, -3) \)
- \( B(6, 2) \)

2. **Let the point M(p, -1) divide the line segment AB in the ratio \( k:1 \).**

3. **Use the section formula to find the coordinates of point M:**
The coordinates of point M that divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \) are given by:
\[ M\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Here, we will consider \( m = k \) and \( n = 1 \).

4. **Substituting the coordinates:**
- For the x-coordinate:
\[ p = \frac{k \cdot 6 + 1 \cdot 1}{k + 1} \]
- For the y-coordinate:
\[ -1 = \frac{k \cdot 2 + 1 \cdot (-3)}{k + 1} \]

5. **Solve the y-coordinate equation:**
\[ -1 = \frac{2k - 3}{k + 1} \]
Cross-multiplying gives:
\[ -1(k + 1) = 2k - 3 \]
Simplifying:
\[ -k - 1 = 2k - 3 \]
Rearranging terms:
\[ -k - 2k = -3 + 1 \]
\[ -3k = -2 \implies k = \frac{2}{3} \]

6. **Substituting k back into the x-coordinate equation:**
\[ p = \frac{\frac{2}{3} \cdot 6 + 1 \cdot 1}{\frac{2}{3} + 1} \]
Simplifying the numerator:
\[ = \frac{4 + 1}{\frac{2}{3} + \frac{3}{3}} = \frac{5}{\frac{5}{3}} = 5 \cdot \frac{3}{5} = 3 \]

7. **Conclusion:**
The point \( M(p, -1) \) divides the line segment in the ratio \( 2:3 \) and the value of \( p \) is \( 3 \).

### Final Answer:
The ratio in which point M divides the line segment is \( 2:3 \) and the value of \( p \) is \( 3 \).