Find the ratio in which the point `(1/2,6)` divides the line segment joining the points (3,5) and (-7,9).

Answer

To find the ratio in which the point \((\frac{1}{2}, 6)\) divides the line segment joining the points \((3, 5)\) and \((-7, 9)\), we will use the section formula.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( A(3, 5) \) and \( B(-7, 9) \) be the two points.
- Let \( P\left(\frac{1}{2}, 6\right) \) be the point that divides the line segment \( AB \).

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

3. **Use the Section Formula**:
- According to the section formula, if a point \( P \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( P \) are given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
- Here, \( m = k \) and \( n = 1 \).

4. **Set Up the Equations**:
- For the x-coordinate:
\[ \frac{k \cdot (-7) + 1 \cdot 3}{k + 1} = \frac{1}{2} \]
- For the y-coordinate:
\[ \frac{k \cdot 9 + 1 \cdot 5}{k + 1} = 6 \]

5. **Solve for k using the x-coordinate**:
- Rearranging the equation:
\[ k \cdot (-7) + 3 = \frac{1}{2}(k + 1) \]
- Multiply both sides by \( 2(k + 1) \):
\[ 2k \cdot (-7) + 6 = k + 1 \]
- Simplifying gives:
\[ -14k + 6 = k + 1 \]
- Rearranging:
\[ -15k = -5 \implies k = \frac{1}{3} \]

6. **Write the Ratio**:
- The ratio in which point \( P \) divides the segment \( AB \) is \( k:1 = \frac{1}{3}:1 \) or \( 1:3 \).

### Final Answer:
The ratio in which the point \((\frac{1}{2}, 6)\) divides the line segment joining the points \((3, 5)\) and \((-7, 9)\) is \( 1:3 \).