To find the ratio in which the line joining the points (3, -4) and (-5, 6) is divided by the x-axis, we can follow these steps:
### Step 1: Identify the Points
We have two points:
- Point A (3, -4)
- Point B (-5, 6)
### Step 2: Determine the Coordinates on the X-axis
The x-axis is defined by the equation y = 0. Therefore, we need to find the coordinates of the point where the line joining A and B intersects the x-axis.
### Step 3: Use the Section Formula
Let the point of intersection on the x-axis be P(x, 0). According to the section formula, if a line segment joining two points (x1, y1) and (x2, y2) is divided by a point in the ratio m:n, the coordinates of the point are given by:
\[
P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
Here, we have:
- \(x_1 = 3\), \(y_1 = -4\)
- \(x_2 = -5\), \(y_2 = 6\)
- The y-coordinate of P is 0 (since it lies on the x-axis).
### Step 4: Set Up the Equation for y-coordinate
Using the section formula for the y-coordinate:
\[
0 = \frac{m \cdot 6 + n \cdot (-4)}{m + n}
\]
### Step 5: Solve for m and n
From the equation above, we can cross-multiply:
\[
0 = m \cdot 6 - n \cdot 4
\]
This simplifies to:
\[
6m = 4n
\]
Dividing both sides by 2 gives:
\[
3m = 2n
\]
### Step 6: Express the Ratio
From the equation \(3m = 2n\), we can express the ratio \(m:n\):
\[
\frac{m}{n} = \frac{2}{3}
\]
Thus, the ratio in which the line joining the points (3, -4) and (-5, 6) is divided by the x-axis is \(2:3\).
### Step 7: Final Ratio
Therefore, the final answer is:
\[
\text{The ratio is } 2:3.
\]
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