What is the ratio in which the point `P((-2)/(5),6)` divides the line joining of A (-4, 3) and B (2, 8)?

Answer

To find the ratio in which the point \( P\left(-\frac{2}{5}, 6\right) \) divides the line segment joining the points \( A(-4, 3) \) and \( B(2, 8) \), we will use the section formula.

### Step-by-Step Solution

1. **Identify the Coordinates:**
- Let \( A(x_1, y_1) = (-4, 3) \)
- Let \( B(x_2, y_2) = (2, 8) \)
- Let \( P(x, y) = \left(-\frac{2}{5}, 6\right) \)

2. **Use the Section Formula:**
The section formula states that if a point \( P \) divides the line segment \( AB \) in the ratio \( m_1 : m_2 \), then:
\[ P_x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2} \]
\[ P_y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \]

3. **Set Up the Equations:**
For the x-coordinate:
\[ -\frac{2}{5} = \frac{m_1 \cdot 2 + m_2 \cdot (-4)}{m_1 + m_2} \]
For the y-coordinate:
\[ 6 = \frac{m_1 \cdot 8 + m_2 \cdot 3}{m_1 + m_2} \]

4. **Cross Multiply the First Equation:**
\[ -\frac{2}{5}(m_1 + m_2) = 2m_1 - 4m_2 \]
This simplifies to:
\[ -\frac{2m_1}{5} - \frac{2m_2}{5} = 2m_1 - 4m_2 \]
Multiplying through by 5 to eliminate the fraction:
\[ -2m_1 - 2m_2 = 10m_1 - 20m_2 \]
Rearranging gives:
\[ 12m_1 = 18m_2 \quad \Rightarrow \quad \frac{m_1}{m_2} = \frac{18}{12} = \frac{3}{2} \]

5. **Cross Multiply the Second Equation:**
\[ 6(m_1 + m_2) = 8m_1 + 3m_2 \]
This simplifies to:
\[ 6m_1 + 6m_2 = 8m_1 + 3m_2 \]
Rearranging gives:
\[ 2m_1 = 3m_2 \quad \Rightarrow \quad \frac{m_1}{m_2} = \frac{3}{2} \]

6. **Conclusion:**
The point \( P \) divides the line segment \( AB \) in the ratio \( 3:2 \).