The xy-plane divides the line joining the points (-1,3,4) aned (2,-5,6)

Options

A. internally in the ratio 2:3
B. externally in the ratio 2:3
C. internally in the ratio 3:2
D. externally in the ratio 3:2

Explanation

To solve the problem of how the XY-plane divides the line segment joining the points (-1, 3, 4) and (2, -5, 6), we will follow these steps:

### Step 1: Identify the coordinates of the points
The points given are:
- Point A: \( P_1 = (-1, 3, 4) \)
- Point B: \( P_2 = (2, -5, 6) \)

### Step 2: Write down the coordinates
From the points, we can identify:
- \( x_1 = -1, y_1 = 3, z_1 = 4 \)
- \( x_2 = 2, y_2 = -5, z_2 = 6 \)

### Step 3: Use the formula for the division of a line segment by a plane
The XY-plane is defined by the equation \( z = 0 \). To find how the XY-plane divides the line segment, we can use the formula:
\[ \text{Ratio} = \frac{-z_1}{z_2} \]
Substituting the values:
\[ \text{Ratio} = \frac{-4}{6} \]

### Step 4: Simplify the ratio
Now, simplifying the ratio:
\[ \text{Ratio} = \frac{-4}{6} = \frac{-2}{3} \]

### Step 5: Interpret the result
Since the result is negative, it indicates that the XY-plane divides the line segment externally. The absolute values of the ratio give us the division:
\[ \text{External Ratio} = 2:3 \]

### Final Answer
The XY-plane divides the line segment joining the points (-1, 3, 4) and (2, -5, 6) externally in the ratio 2:3.

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