Find the ratio in which the line joining the points (1, 3, 2), and (-2, 3, -4) is divided by the
YZ-plane. Also, find the coordinates of the point of division.

Answer

To find the ratio in which the line joining the points \( A(1, 3, 2) \) and \( B(-2, 3, -4) \) 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 \( x = 0 \). Any point on this plane will have its x-coordinate equal to 0.

### Step 2: Use the Section Formula
Let the point of division be \( R(0, y, z) \). According to the section formula, if a point divides the line segment joining two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) in the ratio \( k:1 \), then the coordinates of the point \( R \) can be given by:

\[ R = \left( \frac{k \cdot x_2 + 1 \cdot x_1}{k + 1}, \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1}, \frac{k \cdot z_2 + 1 \cdot z_1}{k + 1} \right) \]

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

\[ 0 = \frac{k \cdot (-2) + 1 \cdot 1}{k + 1} \]

### Step 4: Solve for k
Multiplying both sides by \( (k + 1) \):

\[ 0 = k \cdot (-2) + 1 \]

Rearranging gives:

\[ -2k + 1 = 0 \implies 2k = 1 \implies k = \frac{1}{2} \]

### Step 5: Find the ratio
The ratio in which the line is divided is \( k:1 \), which is \( \frac{1}{2}:1 \) or simplified to \( 1:2 \).

### Step 6: Find the coordinates of the point of division
Now we can find the y and z coordinates of point \( R \) using the section formula:

#### For y-coordinate:
\[ y = \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1} = \frac{\frac{1}{2} \cdot 3 + 1 \cdot 3}{\frac{1}{2} + 1} = \frac{\frac{3}{2} + 3}{\frac{3}{2}} = \frac{\frac{3}{2} + \frac{6}{2}}{\frac{3}{2}} = \frac{\frac{9}{2}}{\frac{3}{2}} = 3 \]

#### For z-coordinate:
\[ z = \frac{k \cdot z_2 + 1 \cdot z_1}{k + 1} = \frac{\frac{1}{2} \cdot (-4) + 1 \cdot 2}{\frac{1}{2} + 1} = \frac{-2 + 2}{\frac{3}{2}} = \frac{0}{\frac{3}{2}} = 0 \]

### Final Result
Thus, the coordinates of the point \( R \) are \( (0, 3, 0) \).

### Summary
- The ratio in which the line joining the points \( (1, 3, 2) \) and \( (-2, 3, -4) \) is divided by the YZ-plane is \( 1:2 \).
- The coordinates of the point of division are \( (0, 3, 0) \).