The ratio in which the point, P(-3,x) divide the line segment joining the points A(-5,-4) and B(-2,3) is :

Options

A. `3:2`
B. `4:7`
C. `2:1`
D. `5:3`

Answer

To solve the problem of finding the ratio in which the point P(-3, x) divides the line segment joining the points A(-5, -4) and B(-2, 3), we can use the section formula. Here’s a step-by-step solution:

### Step 1: Identify the coordinates
Let:
- A = (-5, -4) => (x1, y1)
- B = (-2, 3) => (x2, y2)
- P = (-3, x) => (h, k)

### Step 2: Set up the section formula
According to the section formula, if a point P divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m1:m2, then the coordinates of P can be expressed as:
\[ h = \frac{m1 \cdot x2 + m2 \cdot x1}{m1 + m2} \]
\[ k = \frac{m1 \cdot y2 + m2 \cdot y1}{m1 + m2} \]

### Step 3: Substitute the known values
Substituting the values into the formula for x-coordinate:
\[ -3 = \frac{m1 \cdot (-2) + m2 \cdot (-5)}{m1 + m2} \]

For the y-coordinate:
\[ x = \frac{m1 \cdot 3 + m2 \cdot (-4)}{m1 + m2} \]

### Step 4: Rearranging the equation for x-coordinate
Multiply both sides by (m1 + m2):
\[ -3(m1 + m2) = m1 \cdot (-2) + m2 \cdot (-5) \]
This simplifies to:
\[ -3m1 - 3m2 = -2m1 - 5m2 \]
Rearranging gives:
\[ -3m1 + 2m1 = -5m2 + 3m2 \]
\[ -m1 = -2m2 \]
Thus:
\[ m1 = 2m2 \]

### Step 5: Express the ratio
From the equation \(m1 = 2m2\), we can express the ratio:
\[ \frac{m1}{m2} = \frac{2}{1} \]

### Conclusion
The point P(-3, x) divides the line segment joining A(-5, -4) and B(-2, 3) in the ratio 2:1.