The ratio in which the join of A(1, 2, 3) and B(3, 4, 6) is divided by XY-plane externally is

Options

A. `2 : 1`
B. `1 : 2`
C. `2 : 3`
D. `3 : 2`

Answer

To solve the problem of finding the ratio in which the line segment joining points A(1, 2, 3) and B(3, 4, 6) is divided by the XY-plane externally, we can follow these steps:

### Step 1: Understand the problem
We need to find the ratio in which the line segment AB is divided by the XY-plane. The XY-plane is defined by the equation z = 0.

### Step 2: Set up the coordinates
Let the coordinates of point A be \( A(1, 2, 3) \) and the coordinates of point B be \( B(3, 4, 6) \).

### Step 3: Assume the ratio
Assume the ratio in which the line segment AB is divided by the XY-plane externally is \( k:1 \). This means that if point C divides AB in the ratio \( k:1 \), then we can express the coordinates of point C as:
\[ C\left(\frac{k \cdot x_B + x_A}{k + 1}, \frac{k \cdot y_B + y_A}{k + 1}, \frac{k \cdot z_B + z_A}{k + 1}\right) \]
Where \( (x_A, y_A, z_A) \) and \( (x_B, y_B, z_B) \) are the coordinates of points A and B respectively.

### Step 4: Substitute the coordinates
Substituting the coordinates of A and B into the formula for C:
\[ C\left(\frac{k \cdot 3 + 1}{k + 1}, \frac{k \cdot 4 + 2}{k + 1}, \frac{k \cdot 6 + 3}{k + 1}\right) \]

### Step 5: Set the z-coordinate to zero
Since point C lies on the XY-plane, we set the z-coordinate to zero:
\[ \frac{k \cdot 6 + 3}{k + 1} = 0 \]

### Step 6: Solve for k
To solve for k, we set the numerator equal to zero:
\[ k \cdot 6 + 3 = 0 \]
\[ k \cdot 6 = -3 \]
\[ k = -\frac{1}{2} \]

### Step 7: Determine the external ratio
Since k is negative, it indicates that the division is external. The ratio in which the line segment AB is divided by the XY-plane is:
\[ 1:k = 1:-\frac{1}{2} = 1: \frac{1}{2} \]
Thus, the ratio is \( 2:1 \) when expressed in positive terms.

### Final Answer
The ratio in which the line segment joining A(1, 2, 3) and B(3, 4, 6) is divided by the XY-plane externally is \( 2:1 \).

---