y-axis divides the linje segment joining `(-3,-4)and (1,-2)` in the ratio

Options

A. `1:3`
B. `2:3`
C. `3:1`
D. `3:2`

Answer

To find the ratio in which the y-axis divides the line segment joining the points \((-3, -4)\) and \((1, -2)\), we can follow these steps:

### Step 1: Identify the points
Let the points be:
- \( A(-3, -4) \)
- \( B(1, -2) \)

### Step 2: Assume the point on the y-axis
The y-axis is defined by \( x = 0 \). We can denote the point where the y-axis intersects the line segment joining \( A \) and \( B \) as \( P(0, y) \).

### Step 3: Use the section formula
We assume that point \( P \) divides the line segment \( AB \) in the ratio \( \lambda : 1 \). According to the section formula, the coordinates of point \( P \) can be calculated as follows:

\[ P(x, y) = \left( \frac{n \cdot x_1 + m \cdot x_2}{m + n}, \frac{n \cdot y_1 + m \cdot y_2}{m + n} \right) \]

Here:
- \( (x_1, y_1) = (-3, -4) \)
- \( (x_2, y_2) = (1, -2) \)
- \( m = \lambda \)
- \( n = 1 \)

### Step 4: Set up the equations for x and y coordinates
For the x-coordinate:
\[ 0 = \frac{1 \cdot (-3) + \lambda \cdot (1)}{\lambda + 1} \]
This simplifies to:
\[ 0 = -3 + \lambda \implies \lambda = 3 \]

For the y-coordinate:
\[ y = \frac{1 \cdot (-4) + \lambda \cdot (-2)}{\lambda + 1} \]
Substituting \( \lambda = 3 \):
\[ y = \frac{1 \cdot (-4) + 3 \cdot (-2)}{3 + 1} = \frac{-4 - 6}{4} = \frac{-10}{4} = -\frac{5}{2} \]

### Step 5: Conclusion
Thus, the point \( P \) is \( (0, -\frac{5}{2}) \) and the ratio in which the y-axis divides the line segment \( AB \) is \( 3:1 \).

### Final Answer
The y-axis divides the line segment joining the points \((-3, -4)\) and \((1, -2)\) in the ratio \( 3:1 \).

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