In what ratio does the point P (a, 2) divide the line segment joining the points A(5,-3) and B(-9, 4) ? Also, find the value of 'a'.

Answer

To solve the problem, we need to determine the ratio in which the point P(a, 2) divides the line segment joining the points A(5, -3) and B(-9, 4). We will also find the value of 'a'.

### Step-by-Step Solution:

1. **Identify the Coordinates:**
- Point A has coordinates (5, -3).
- Point B has coordinates (-9, 4).
- Point P has coordinates (a, 2).

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 point P can be given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Here, we will denote the ratio as λ:1, where λ is the ratio in which point P divides the line segment.

3. **Set Up the Equation for Y-Coordinates:**
Using the y-coordinates of points A and B:
\[ 2 = \frac{4\lambda + (-3) \cdot 1}{\lambda + 1} \]
Simplifying this equation:
\[ 2(\lambda + 1) = 4\lambda - 3 \]
\[ 2\lambda + 2 = 4\lambda - 3 \]
Rearranging gives:
\[ 4\lambda - 2\lambda = 5 \]
\[ 2\lambda = 5 \implies \lambda = \frac{5}{2} \]

4. **Determine the Ratio:**
The ratio in which point P divides the line segment AB is:
\[ \text{Ratio} = \lambda:1 = \frac{5}{2}:1 = 5:2 \]

5. **Set Up the Equation for X-Coordinates:**
Now we will use the x-coordinates of points A and B:
\[ a = \frac{-9\lambda + 5 \cdot 1}{\lambda + 1} \]
Substituting λ = \(\frac{5}{2}\):
\[ a = \frac{-9 \cdot \frac{5}{2} + 5}{\frac{5}{2} + 1} \]
Simplifying:
\[ a = \frac{-\frac{45}{2} + 5}{\frac{5}{2} + 1} = \frac{-\frac{45}{2} + \frac{10}{2}}{\frac{5}{2} + \frac{2}{2}} = \frac{-\frac{35}{2}}{\frac{7}{2}} \]
\[ a = \frac{-35}{7} = -5 \]

6. **Final Results:**
- The ratio in which point P divides the line segment AB is \(5:2\).
- The value of 'a' is \(-5\).

### Summary:
- The point P(a, 2) divides the line segment joining A(5, -3) and B(-9, 4) in the ratio **5:2**.
- The value of 'a' is **-5**.