Find the ratio in which the x-axis divides the segment joining (-3, -5) and (-1, 1).

Answer

To find the ratio in which the x-axis divides the segment joining the points (-3, -5) and (-1, 1), we can follow these steps:

### Step 1: Identify the Points
Let point P = (-3, -5) and point Q = (-1, 1).

### Step 2: Determine the Coordinates of the Intersection Point
Since the x-axis is being intersected, the y-coordinate of the intersection point R will be 0. Therefore, we can express the coordinates of point R as (x, 0).

### Step 3: Use the Section Formula
The section formula states that if a point R divides the line segment joining points P(x1, y1) and Q(x2, y2) in the ratio m1:m2, then the coordinates of R can be expressed as:
\[ R = \left( \frac{m1 \cdot x2 + m2 \cdot x1}{m1 + m2}, \frac{m1 \cdot y2 + m2 \cdot y1}{m1 + m2} \right) \]
Here, we know that the y-coordinate of R is 0, so we can set up the equation:
\[ 0 = \frac{m1 \cdot y2 + m2 \cdot y1}{m1 + m2} \]

### Step 4: Substitute the Known Values
Substituting the known values:
- \(y1 = -5\) (from point P)
- \(y2 = 1\) (from point Q)

We get:
\[ 0 = \frac{m1 \cdot 1 + m2 \cdot (-5)}{m1 + m2} \]

### Step 5: Simplify the Equation
Cross-multiplying gives us:
\[ 0 = m1 - 5m2 \]
This implies:
\[ m1 = 5m2 \]

### Step 6: Find the Ratio
From the equation \(m1 = 5m2\), we can express the ratio \(m1:m2\) as:
\[ \frac{m1}{m2} = 5 \]
Thus, the ratio in which the x-axis divides the segment joining the points (-3, -5) and (-1, 1) is 5:1.

### Final Answer
The x-axis divides the segment in the ratio of 5:1.

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