In what ratio do the coordinate axes divide the line segment joing (-2,5) and (3,-4).

Answer

To find the ratio in which the coordinate axes divide the line segment joining the points A(-2, 5) and B(3, -4), we will solve for both the x-axis and the y-axis separately.

### Step 1: Finding the ratio with respect to the x-axis

1. **Identify the points**: Let A = (-2, 5) and B = (3, -4).
2. **Assume the point of intersection with the x-axis**: Let the point where the line segment AB intersects the x-axis be P, which has coordinates (x, 0).
3. **Use the section formula**: If the ratio in which the x-axis divides the segment AB is k:1, then using the section formula, the coordinates of point P can be expressed as:
\[ P = \left( \frac{3k + (-2)}{k + 1}, \frac{-4k + 5}{k + 1} \right) \]
4. **Set the y-coordinate to 0**: Since P lies on the x-axis, we set the y-coordinate to 0:
\[ \frac{-4k + 5}{k + 1} = 0 \]
5. **Solve for k**:
\[ -4k + 5 = 0 \implies 4k = 5 \implies k = \frac{5}{4} \]

### Step 2: Finding the ratio with respect to the y-axis

1. **Assume the point of intersection with the y-axis**: Let the point where the line segment AB intersects the y-axis be Q, which has coordinates (0, y).
2. **Use the section formula**: If the ratio in which the y-axis divides the segment AB is k:1, then the coordinates of point Q can be expressed as:
\[ Q = \left( \frac{3k + (-2)}{k + 1}, \frac{-4k + 5}{k + 1} \right) \]
3. **Set the x-coordinate to 0**: Since Q lies on the y-axis, we set the x-coordinate to 0:
\[ \frac{3k - 2}{k + 1} = 0 \]
4. **Solve for k**:
\[ 3k - 2 = 0 \implies 3k = 2 \implies k = \frac{2}{3} \]

### Final Answer

The ratio in which the coordinate axes divide the line segment joining the points (-2, 5) and (3, -4) is:
- For the x-axis: \( \frac{5}{4} : 1 \)
- For the y-axis: \( \frac{2}{3} : 1 \)