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

Answer

To solve the problem of finding the ratio in which point P(4, p) divides the line segment joining points A(2, 3) and B(6, 3), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Coordinates:**
- Let A = (2, 3) and B = (6, 3).
- Point P has coordinates (4, p).

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 m1:m2, then:
\[ x = \frac{m1 \cdot x2 + m2 \cdot x1}{m1 + m2} \]
\[ y = \frac{m1 \cdot y2 + m2 \cdot y1}{m1 + m2} \]

3. **Set Up the Equation for x-coordinate:**
Since we know the x-coordinate of point P is 4, we can write:
\[ 4 = \frac{m1 \cdot 6 + m2 \cdot 2}{m1 + m2} \]

4. **Cross Multiply to Eliminate the Denominator:**
\[ 4(m1 + m2) = m1 \cdot 6 + m2 \cdot 2 \]
Expanding this gives:
\[ 4m1 + 4m2 = 6m1 + 2m2 \]

5. **Rearrange the Equation:**
Rearranging the terms:
\[ 4m1 + 4m2 - 6m1 - 2m2 = 0 \]
This simplifies to:
\[ -2m1 + 2m2 = 0 \]
or
\[ 2m2 = 2m1 \implies m1 = m2 \]

6. **Determine the Ratio:**
Since \( m1 = m2 \), we can conclude that the ratio is:
\[ m1:m2 = 1:1 \]

7. **Use the Section Formula for y-coordinate:**
Now, we will find the value of p using the y-coordinates. Since the ratio is 1:1, we can use:
\[ p = \frac{m1 \cdot y2 + m2 \cdot y1}{m1 + m2} \]
Substituting the values:
\[ p = \frac{1 \cdot 3 + 1 \cdot 3}{1 + 1} = \frac{3 + 3}{2} = \frac{6}{2} = 3 \]

8. **Final Result:**
The value of p is 3, and the ratio in which P divides the segment AB is 1:1.