The ratio in which the line joining (2,4,5) and (3,5,-4) is divided by the yz-plane is

Options

A. 2:3
B. 3:2
C. -2:3
D. 4:(-3)

Answer

To find the ratio in which the line joining the points \( A(2, 4, 5) \) and \( B(3, 5, -4) \) is divided by the YZ-plane, we can follow these steps:

### Step 1: Understand the YZ-plane
The YZ-plane is defined by the equation \( x = 0 \). This means that any point on the YZ-plane has an x-coordinate of 0.

### Step 2: Use the Section Formula
The section formula states that if a point \( P(x, y, z) \) divides the line segment joining points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) in the ratio \( \lambda : 1 \), then the coordinates of point \( P \) can be expressed as:
\[ P\left( \frac{\lambda x_2 + x_1}{\lambda + 1}, \frac{\lambda y_2 + y_1}{\lambda + 1}, \frac{\lambda z_2 + z_1}{\lambda + 1} \right) \]

### Step 3: Set up the Equation for the X-coordinate
Since we want to find the intersection with the YZ-plane, we set the x-coordinate of point \( P \) to 0:
\[ \frac{\lambda x_2 + x_1}{\lambda + 1} = 0 \]
Substituting \( x_1 = 2 \) and \( x_2 = 3 \):
\[ \frac{\lambda \cdot 3 + 2}{\lambda + 1} = 0 \]

### Step 4: Solve the Equation
To solve the equation, we set the numerator equal to zero:
\[ \lambda \cdot 3 + 2 = 0 \]
\[ \lambda \cdot 3 = -2 \]
\[ \lambda = -\frac{2}{3} \]

### Step 5: Interpret the Ratio
The ratio in which the line is divided by the YZ-plane is \( \lambda : 1 \), which translates to:
\[ -\frac{2}{3} : 1 \]
This can be expressed as \( -2 : 3 \).

### Conclusion
Thus, the ratio in which the line joining the points \( (2, 4, 5) \) and \( (3, 5, -4) \) is divided by the YZ-plane is:
\[ \text{Ratio} = -2 : 3 \]