The line x = 0 divides the line joining the points (3 , -5) and (-4 , 7) in the ratio

Options

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

Answer

To find the ratio in which the line \( x = 0 \) divides the line segment joining the points \( (3, -5) \) and \( (-4, 7) \), we can use the section formula. Here’s how to solve it step by step:

### Step 1: Identify the points
Let the points be:
- \( A(3, -5) \) (Point 1)
- \( B(-4, 7) \) (Point 2)

### Step 2: Set up the section formula
The section formula states that if a point \( P \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( P \) can be expressed as:
\[ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
In our case, since the line \( x = 0 \) divides the segment, the x-coordinate of point \( P \) will be 0.

### Step 3: Set up the equation for the x-coordinate
Using the section formula for the x-coordinate:
\[ 0 = \frac{m(-4) + n(3)}{m+n} \]
This implies:
\[ m(-4) + n(3) = 0 \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ 3n = 4m \]
This can be rewritten as:
\[ \frac{m}{n} = \frac{3}{4} \]
Thus, \( m:n = 3:4 \).

### Step 5: Conclusion
The line \( x = 0 \) divides the line segment joining the points \( (3, -5) \) and \( (-4, 7) \) in the ratio \( 3:4 \).

### Final Answer
The ratio in which the line \( x = 0 \) divides the line segment joining the points \( (3, -5) \) and \( (-4, 7) \) is \( 3:4 \).

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