Find the ratio in which Y-axis divides the line segment joining the points (-1,-4) and (5,-6). Also find the coordinates of the point of the intersection.

Answer

To find the ratio in which the Y-axis divides the line segment joining the points (-1, -4) and (5, -6), and to find the coordinates of the point of intersection, we can follow these steps:

### Step 1: Identify the Points
Let the two points be:
- Point A: (-1, -4) (x1, y1)
- Point B: (5, -6) (x2, y2)

### Step 2: Use the Section Formula
The Y-axis divides the line segment in the ratio of \( \lambda : 1 \). According to the section formula, the coordinates of the point dividing the line segment in the ratio \( m:n \) are given by:

\[ x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} \]
\[ y = \frac{m \cdot y_2 + n \cdot y_1}{m + n} \]

In our case, \( m = \lambda \) and \( n = 1 \).

### Step 3: Set the x-coordinate to 0
Since the point lies on the Y-axis, the x-coordinate must be 0. Therefore, we set up the equation:

\[ 0 = \frac{\lambda \cdot 5 + 1 \cdot (-1)}{\lambda + 1} \]

### Step 4: Solve for \( \lambda \)
To find \( \lambda \), we can solve the equation:

\[ 0 = \lambda \cdot 5 - 1 \]

This simplifies to:

\[ \lambda \cdot 5 = 1 \]
\[ \lambda = \frac{1}{5} \]

### Step 5: Determine the Ratio
The ratio in which the Y-axis divides the line segment is:

\[ \text{Ratio} = \lambda : 1 = \frac{1}{5} : 1 = 1 : 5 \]

### Step 6: Find the y-coordinate
Now, we can find the y-coordinate of the point of intersection using the section formula:

\[ y = \frac{\lambda \cdot y_2 + 1 \cdot y_1}{\lambda + 1} \]

Substituting the values:

\[ y = \frac{\frac{1}{5} \cdot (-6) + 1 \cdot (-4)}{\frac{1}{5} + 1} \]
\[ y = \frac{-\frac{6}{5} - 4}{\frac{1}{5} + 1} \]
\[ y = \frac{-\frac{6}{5} - \frac{20}{5}}{\frac{6}{5}} = \frac{-\frac{26}{5}}{\frac{6}{5}} = -\frac{26}{6} = -\frac{13}{3} \]

### Step 7: Final Coordinates
Thus, the coordinates of the point of intersection are:

\[ (0, -\frac{13}{3}) \]

### Summary
- The ratio in which the Y-axis divides the line segment is \( 1 : 5 \).
- The coordinates of the point of intersection are \( (0, -\frac{13}{3}) \).

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