Calculate the ratio in which the line joining A (-4, 2) and B (3, 6) is divided by point P (x, 3). Also, find (i) x

Answer

To solve the problem, we need to find the ratio in which the line segment joining points A (-4, 2) and B (3, 6) is divided by the point P (x, 3). We will also find the value of x.

### Step-by-Step Solution

1. **Identify the Coordinates:**
- Let A = (-4, 2)
- Let B = (3, 6)
- Let P = (x, 3)

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 m:n, then the coordinates of P are given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

3. **Set Up the Equation for y-coordinate:**
We know the y-coordinate of point P is 3. Therefore, we can set up the equation using the y-coordinates of A and B:
\[ 3 = \frac{m \cdot 6 + n \cdot 2}{m+n} \]

4. **Express the Ratio:**
Let the ratio be \( k:1 \) (where \( m = k \) and \( n = 1 \)). Substitute into the equation:
\[ 3 = \frac{k \cdot 6 + 1 \cdot 2}{k + 1} \]

5. **Cross Multiply:**
Cross multiplying gives:
\[ 3(k + 1) = 6k + 2 \]
Expanding this:
\[ 3k + 3 = 6k + 2 \]

6. **Rearranging the Equation:**
Rearranging gives:
\[ 3k - 6k = 2 - 3 \]
Simplifying:
\[ -3k = -1 \]
Therefore:
\[ k = \frac{1}{3} \]

7. **Determine the Ratio:**
The ratio in which P divides AB is \( k:1 = \frac{1}{3}:1 \) or \( 1:3 \).

8. **Set Up the Equation for x-coordinate:**
Now we will find the x-coordinate using the section formula:
\[ x = \frac{k \cdot x_2 + n \cdot x_1}{k+n} \]
Substituting the values:
\[ x = \frac{\frac{1}{3} \cdot 3 + 1 \cdot (-4)}{\frac{1}{3} + 1} \]

9. **Calculate the x-coordinate:**
Simplifying the numerator:
\[ x = \frac{1 + (-4)}{\frac{1}{3} + 1} = \frac{-3}{\frac{1}{3} + \frac{3}{3}} = \frac{-3}{\frac{4}{3}} = -3 \cdot \frac{3}{4} = -\frac{9}{4} \]

### Final Answer:
- The ratio in which the line joining A and B is divided by point P is **1:3**.
- The value of **x** is **-9/4**.