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 \( P(3, 0, 5) \) and \( Q(-2, 3, 2) \) is divided by the YZ-plane, we need to determine where this line segment intersects the YZ-plane. The equation of the YZ-plane is given by \( x = 0 \).

### Step 1: Understand the Section Formula
The section formula states that if a point \( R \) divides the line segment joining points \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) in the ratio \( m:n \), then the coordinates of point \( R \) can be given as:
\[ R = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]

### Step 2: Set Up the Problem
Given:
- \( P(3, 0, 5) \) where \( x_1 = 3, y_1 = 0, z_1 = 5 \)
- \( Q(-2, 3, 2) \) where \( x_2 = -2, y_2 = 3, z_2 = 2 \)

We want to find the point \( R \) where \( x = 0 \).

### Step 3: Apply the Section Formula
Using the section formula, we set \( R \) as:
\[ R = \left( \frac{m(-2) + n(3)}{m+n}, \frac{m(3) + n(0)}{m+n}, \frac{m(2) + n(5)}{m+n} \right) \]

Since we want \( R \) to lie on the YZ-plane, we set the x-coordinate to 0:
\[ \frac{m(-2) + n(3)}{m+n} = 0 \]

### Step 4: Solve for the Ratio
From the equation above, we can cross-multiply:
\[ m(-2) + n(3) = 0 \]
This simplifies to:
\[ 3n = 2m \quad \Rightarrow \quad \frac{m}{n} = \frac{3}{2} \]

Thus, the ratio \( m:n \) is \( 3:2 \).

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

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