In which ratio the line segment joining the points (3, 0, 5) and (-2, 3, 2) is divided by YZ-plane?

Options

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

Answer

To find the ratio in which the line segment joining the points \( A(3, 0, 5) \) and \( B(-2, 3, 2) \) 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 will have an x-coordinate of 0.

### Step 2: Use the Section Formula
Let the point where the line segment \( AB \) intersects the YZ-plane be \( R \). If the ratio in which the point \( R \) divides the line segment \( AB \) is \( \lambda : 1 \), then according to the section formula, the coordinates of point \( R \) can be given as:

\[ R = \left( \frac{-2\lambda + 3}{\lambda + 1}, \frac{3\lambda + 0}{\lambda + 1}, \frac{2\lambda + 5}{\lambda + 1} \right) \]

### Step 3: Set the x-coordinate to 0
Since point \( R \) lies on the YZ-plane, we set the x-coordinate to 0:

\[ \frac{-2\lambda + 3}{\lambda + 1} = 0 \]

### Step 4: Solve for \( \lambda \)
To solve for \( \lambda \), we multiply both sides by \( \lambda + 1 \) (assuming \( \lambda + 1 \neq 0 \)):

\[ -2\lambda + 3 = 0 \]

Rearranging gives:

\[ 2\lambda = 3 \quad \Rightarrow \quad \lambda = \frac{3}{2} \]

### Step 5: Find the ratio
The ratio in which the point divides the line segment \( AB \) is \( \lambda : 1 \), which translates to:

\[ \frac{3}{2} : 1 = 3 : 2 \]

### Conclusion
Thus, the line segment joining the points \( (3, 0, 5) \) and \( (-2, 3, 2) \) is divided by the YZ-plane in the ratio \( 3 : 2 \).