In what ratio is the line segment made by the points (7, 3) and (- 4, 5) divided by the y-axis?

Options

A. `2 : 3`
B. `4 : 7`
C. `3 : 5`
D. `7 : 4`

Answer

To find the ratio in which the line segment joining the points \( A(7, 3) \) and \( B(-4, 5) \) is divided by the y-axis, we can follow these steps:

### Step 1: Identify the points
Let \( A(7, 3) \) and \( B(-4, 5) \) be the two points.

### Step 2: Understand the division by the y-axis
The y-axis is represented by the line \( x = 0 \). We need to find the point \( P \) on the y-axis that divides the line segment \( AB \).

### Step 3: Use the section formula
The coordinates of point \( P \) that divides the line segment \( AB \) in the ratio \( m:n \) can be calculated using the section formula:
\[ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
Here, \( (x_1, y_1) = (7, 3) \) and \( (x_2, y_2) = (-4, 5) \).

### Step 4: Set the x-coordinate to 0
Since point \( P \) lies on the y-axis, its x-coordinate is 0. Therefore, we set up the equation:
\[ 0 = \frac{m(-4) + n(7)}{m+n} \]
This implies:
\[ m(-4) + n(7) = 0 \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ 7n = 4m \quad \Rightarrow \quad \frac{m}{n} = \frac{7}{4} \]

### Step 6: Express the ratio
This means that the ratio \( m:n \) is \( 7:4 \).

### Conclusion
Thus, the y-axis divides the line segment joining the points \( (7, 3) \) and \( (-4, 5) \) in the ratio \( 7:4 \).

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