In what ratio does the point `P(2,(-5)/(6))` divide the line joining the points `A(-3,5)` and `B(3,-2)` ?

Options

A. `1:5`
B. `5:1`
C. `2:3`
D. `3:5`

Answer

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

### Step-by-Step Solution:

1. **Identify the Coordinates:**
- Let \( A(-3, 5) \) be \( (x_1, y_1) \)
- Let \( B(3, -2) \) be \( (x_2, y_2) \)
- Let \( P(2, -\frac{5}{6}) \) be the point that divides the line segment \( AB \).

2. **Assume the Ratio:**
- Let the ratio in which \( P \) divides \( AB \) be \( \lambda : 1 \).

3. **Use the Section Formula:**
- The coordinates of point \( P \) can be expressed using the section formula:
\[ P\left( \frac{x_1 \cdot n + x_2 \cdot m}{m+n}, \frac{y_1 \cdot n + y_2 \cdot m}{m+n} \right) \]
- Here, \( m = \lambda \) and \( n = 1 \).

4. **Set Up the Equations:**
- For the x-coordinate:
\[ 2 = \frac{-3 \cdot 1 + 3 \cdot \lambda}{\lambda + 1} \]
- For the y-coordinate:
\[ -\frac{5}{6} = \frac{5 \cdot 1 + (-2) \cdot \lambda}{\lambda + 1} \]

5. **Solve the x-coordinate Equation:**
- Cross-multiply:
\[ 2(\lambda + 1) = -3 + 3\lambda \]
- Expand:
\[ 2\lambda + 2 = -3 + 3\lambda \]
- Rearranging gives:
\[ 2 + 3 = 3\lambda - 2\lambda \]
\[ 5 = \lambda \]

6. **Conclusion:**
- The ratio in which point \( P \) divides the line segment \( AB \) is \( \lambda : 1 = 5 : 1 \).

### Final Answer:
The point \( P(2, -\frac{5}{6}) \) divides the line segment joining \( A(-3, 5) \) and \( B(3, -2) \) in the ratio \( 5 : 1 \).