Find the ratio in which P(4, m) divides the line segment joining the points A(2, 3) and B(6, 3). Hence find m.

Answer

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

### Step-by-Step Solution:

1. **Identify the Coordinates:**
- Let A(2, 3) and B(6, 3).
- Let P(4, m) be the point that divides the line segment AB.

2. **Use the Section Formula:**
The section formula states that if a point P divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio k:1, then the coordinates of point P can be calculated as:
\[ P\left(\frac{kx_2 + x_1}{k + 1}, \frac{ky_2 + y_1}{k + 1}\right) \]
Here, we need to find the ratio k in which P divides AB.

3. **Find the x-coordinate:**
- The x-coordinate of point P is given as 4.
- Using the section formula for the x-coordinate:
\[ 4 = \frac{k \cdot 6 + 2}{k + 1} \]
- Cross-multiplying gives:
\[ 4(k + 1) = 6k + 2 \]
- Expanding and simplifying:
\[ 4k + 4 = 6k + 2 \]
\[ 4k - 6k = 2 - 4 \]
\[ -2k = -2 \implies k = 1 \]

4. **Find the y-coordinate:**
- Now, we need to find the value of m using the y-coordinate.
- The y-coordinate of point P is m, and we can use the section formula for the y-coordinate:
\[ m = \frac{1 \cdot 3 + 3}{1 + 1} = \frac{3 + 3}{2} = \frac{6}{2} = 3 \]

5. **Conclusion:**
- The ratio in which P divides the line segment AB is 1:1.
- The value of m is 3.

### Final Answer:
- The ratio is 1:1.
- The value of m is 3.