The line y = 0 divides the line joining the points (3, -5) and (- 4, 7) in the ratio
(a)`3 : 4`
(b)`4 : 5`
(c)`5 : 7`
(d)`7 : 9`

Options

A. `3 : 4`
B. `4 : 5`
C. `5 : 7`
D. `7 : 9`

Answer

To solve the problem of finding the ratio in which the line \( y = 0 \) divides the line segment joining the points \( (3, -5) \) and \( (-4, 7) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points**:
We have two points:
- Point 1: \( (x_1, y_1) = (3, -5) \)
- Point 2: \( (x_2, y_2) = (-4, 7) \)

2. **Use the Section Formula**:
The section formula states that if a line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is divided by a point \( P(x, y) \) in the ratio \( m:n \), then:
\[ y = \frac{m \cdot y_2 + n \cdot y_1}{m + n} \]
In our case, we want to find the ratio \( m:n \) such that \( y = 0 \).

3. **Set Up the Equation**:
Let’s assume the ratio in which the line divides the segment is \( k:1 \) (where \( k \) is the unknown we need to find). Thus, we can write:
\[ 0 = \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1} \]
Substituting \( y_1 = -5 \) and \( y_2 = 7 \):
\[ 0 = \frac{k \cdot 7 + 1 \cdot (-5)}{k + 1} \]

4. **Simplify the Equation**:
This simplifies to:
\[ 0 = \frac{7k - 5}{k + 1} \]
For this fraction to be zero, the numerator must be zero:
\[ 7k - 5 = 0 \]

5. **Solve for \( k \)**:
Rearranging gives:
\[ 7k = 5 \implies k = \frac{5}{7} \]

6. **Determine the Ratio**:
The ratio \( m:n \) is \( k:1 \), which translates to:
\[ \frac{5}{7} : 1 \]
This can be expressed as:
\[ 5 : 7 \]

### Conclusion:
The line \( y = 0 \) divides the line segment joining the points \( (3, -5) \) and \( (-4, 7) \) in the ratio \( 5:7 \). Therefore, the correct option is **(c) 5 : 7**.