If the point P(6,2) divides the line segment joining A(6,5) and B(4, y) in the ratio 3:1, then find the value of y.

Answer

To solve the problem of finding the value of \( y \) such that the point \( P(6, 2) \) divides the line segment joining \( A(6, 5) \) and \( B(4, y) \) in the ratio \( 3:1 \), we can use the section formula.

### Step-by-step Solution:

1. **Understanding the Section Formula**:
The section formula states that if a point \( P(x, y) \) divides the line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( P \) can be calculated as:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

2. **Identifying the Coordinates**:
Here, we have:
- \( A(6, 5) \) with coordinates \( (x_1, y_1) = (6, 5) \)
- \( B(4, y) \) with coordinates \( (x_2, y_2) = (4, y) \)
- The ratio \( m:n = 3:1 \)
- The coordinates of point \( P \) are \( (6, 2) \)

3. **Setting Up the Equation for x-coordinate**:
Using the section formula for the x-coordinate:
\[ 6 = \frac{3 \cdot 4 + 1 \cdot 6}{3 + 1} \]
Simplifying the right side:
\[ 6 = \frac{12 + 6}{4} = \frac{18}{4} = 4.5 \]
Since \( 6 \neq 4.5 \), we need to check the y-coordinate.

4. **Setting Up the Equation for y-coordinate**:
Using the section formula for the y-coordinate:
\[ 2 = \frac{3y + 1 \cdot 5}{3 + 1} \]
Simplifying the right side:
\[ 2 = \frac{3y + 5}{4} \]
Multiplying both sides by 4 to eliminate the fraction:
\[ 8 = 3y + 5 \]

5. **Solving for y**:
Rearranging the equation:
\[ 3y = 8 - 5 \]
\[ 3y = 3 \]
Dividing both sides by 3:
\[ y = 1 \]

### Final Answer:
The value of \( y \) is \( 1 \).