Ratio in which the zx-plane divides the join of `(1, 2, 3) and (4, 2, 1)`.

Options

A. 1:1 internally
B. 1:1 externally
C. 2:1 internally
D. 2:1 externally

Answer

To find the ratio in which the zx-plane divides the line segment joining the points \( A(1, 2, 3) \) and \( B(4, 2, 1) \), we can follow these steps:

### Step 1: Understand the zx-plane
The zx-plane is defined by the equation \( y = 0 \). This means that any point on this plane will have a y-coordinate of 0.

### Step 2: Set up the section formula
Let the point \( P \) divide the line segment \( AB \) in the ratio \( k:1 \). The coordinates of point \( P \) can be expressed using the section formula:
\[ P = \left( \frac{k \cdot x_2 + 1 \cdot x_1}{k + 1}, \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1}, \frac{k \cdot z_2 + 1 \cdot z_1}{k + 1} \right) \]
where \( A(x_1, y_1, z_1) = (1, 2, 3) \) and \( B(x_2, y_2, z_2) = (4, 2, 1) \).

### Step 3: Substitute the coordinates
Substituting the coordinates of points \( A \) and \( B \) into the section formula gives:
\[ P = \left( \frac{k \cdot 4 + 1 \cdot 1}{k + 1}, \frac{k \cdot 2 + 1 \cdot 2}{k + 1}, \frac{k \cdot 1 + 1 \cdot 3}{k + 1} \right) \]
This simplifies to:
\[ P = \left( \frac{4k + 1}{k + 1}, \frac{2k + 2}{k + 1}, \frac{k + 3}{k + 1} \right) \]

### Step 4: Set the y-coordinate to 0
Since \( P \) lies on the zx-plane, we set the y-coordinate to 0:
\[ \frac{2k + 2}{k + 1} = 0 \]

### Step 5: Solve for \( k \)
To solve for \( k \), we set the numerator equal to zero:
\[ 2k + 2 = 0 \implies 2k = -2 \implies k = -1 \]

### Step 6: Determine the ratio
The ratio in which the zx-plane divides the line segment \( AB \) is \( k:1 \), which is:
\[ -1:1 \]
This indicates that the zx-plane divides the segment externally.

### Final Answer
Thus, the ratio in which the zx-plane divides the join of the points \( (1, 2, 3) \) and \( (4, 2, 1) \) is:
\[ \text{Ratio} = 1:1 \text{ externally} \]

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