Find the ratio in which the line joining (– 4, 3) and (5, –2) is divided by (i) x-axis (ii) y-axis.
(a)4:5 and 3:4
(b)5:6 and 3:6
(c) 3:6 and 1:5
(d) None of these

Answer

To solve the problem of finding the ratio in which the line joining the points (-4, 3) and (5, -2) is divided by the x-axis and the y-axis, we will follow these steps:

### Step 1: Finding the ratio for the x-axis

1. **Identify the coordinates**: The points are A(-4, 3) and B(5, -2).
2. **Set the y-coordinate to 0**: Since we are finding where the line intersects the x-axis, we set y = 0.
3. **Use the section formula**: The section formula states that if a point P divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m1:m2, then:
\[ y = \frac{m2 \cdot y1 + m1 \cdot y2}{m1 + m2} \]
Here, we let the ratio be k:1 (where m1 = k and m2 = 1).

4. **Substituting the values**:
\[ 0 = \frac{1 \cdot 3 + k \cdot (-2)}{k + 1} \]
This simplifies to:
\[ 0 = 3 - 2k \]
Rearranging gives:
\[ 2k = 3 \quad \Rightarrow \quad k = \frac{3}{2} \]

5. **Expressing the ratio**: Thus, the ratio in which the line joins the points (-4, 3) and (5, -2) is divided by the x-axis is:
\[ 3:2 \]

### Step 2: Finding the ratio for the y-axis

1. **Set the x-coordinate to 0**: Since we are finding where the line intersects the y-axis, we set x = 0.
2. **Using the section formula again**: The formula for the x-coordinate is:
\[ x = \frac{m2 \cdot x1 + m1 \cdot x2}{m1 + m2} \]
Here, we let the ratio be k:1 (where m1 = k and m2 = 1).

3. **Substituting the values**:
\[ 0 = \frac{1 \cdot (-4) + k \cdot 5}{k + 1} \]
This simplifies to:
\[ 0 = -4 + 5k \]
Rearranging gives:
\[ 5k = 4 \quad \Rightarrow \quad k = \frac{4}{5} \]

4. **Expressing the ratio**: Thus, the ratio in which the line joins the points (-4, 3) and (5, -2) is divided by the y-axis is:
\[ 4:5 \]

### Final Result

The ratios are:
- For the x-axis: **3:2**
- For the y-axis: **4:5**

### Conclusion

Since the answer is not listed in the options provided, the correct choice is:
**(d) None of these.**

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