Find the ratio in which the point `P(m, 6)` divides the join of `A(-4, 3)` and `B(2, 8)`. Also, find the value of `m`.

Options

A. `(-2)/(5)`
B. `2/5`
C. `-2`
D. Nome of these

Explanation

To solve the problem of finding the ratio in which the point \( P(m, 6) \) divides the line segment joining points \( A(-4, 3) \) and \( B(2, 8) \), we can use the section formula.

### Step-by-Step Solution:

1. **Identify the Coordinates**:
- Let \( A(-4, 3) \) and \( B(2, 8) \).
- The coordinates of point \( P \) are \( (m, 6) \).

2. **Use the Section Formula**:
The section formula states that if a point \( P(x, y) \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( k_1:k_2 \), then:
\[ x = \frac{k_1 x_2 + k_2 x_1}{k_1 + k_2} \]
\[ y = \frac{k_1 y_2 + k_2 y_1}{k_1 + k_2} \]

3. **Set Up the Equations**:
Let the ratio in which \( P \) divides \( AB \) be \( k:1 \). Then, we can express the coordinates of \( P \) as:
\[ m = \frac{k \cdot 2 + 1 \cdot (-4)}{k + 1} \]
\[ 6 = \frac{k \cdot 8 + 1 \cdot 3}{k + 1} \]

4. **Solve for \( k \)**:
Start with the equation for \( y \):
\[ 6 = \frac{8k + 3}{k + 1} \]
Cross-multiplying gives:
\[ 6(k + 1) = 8k + 3 \]
Expanding and simplifying:
\[ 6k + 6 = 8k + 3 \]
\[ 6 - 3 = 8k - 6k \]
\[ 3 = 2k \implies k = \frac{3}{2} \]

5. **Substitute \( k \) to Find \( m \)**:
Now substitute \( k = \frac{3}{2} \) into the equation for \( x \):
\[ m = \frac{\frac{3}{2} \cdot 2 + 1 \cdot (-4)}{\frac{3}{2} + 1} \]
Simplifying:
\[ m = \frac{3 - 4}{\frac{3}{2} + 1} = \frac{-1}{\frac{5}{2}} = -\frac{2}{5} \]

6. **Final Answer**:
The ratio in which point \( P \) divides the line segment \( AB \) is \( \frac{3}{2} : 1 \) and the value of \( m \) is \( -\frac{2}{5} \).