In which ratio the line segment joining the points
(2, 4, 5) and (3, -4, -5) is divided by the XY-plane?

Options

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

Answer

To find the ratio in which the line segment joining the points \( A(2, 4, 5) \) and \( B(3, -4, -5) \) is divided by the XY-plane, we can follow these steps:

### Step 1: Understand the problem
We need to determine the point \( P \) that divides the line segment \( AB \) in the ratio \( k:1 \) and lies on the XY-plane. The XY-plane is defined by the equation \( z = 0 \).

### Step 2: Set up the coordinates of point \( P \)
Let \( P \) divide the 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 \( A(2, 4, 5) \) corresponds to \( (x_1, y_1, z_1) \) and \( B(3, -4, -5) \) corresponds to \( (x_2, y_2, z_2) \).

### Step 3: Substitute the coordinates
Substituting the coordinates of points \( A \) and \( B \):
\[ P = \left( \frac{k \cdot 3 + 1 \cdot 2}{k + 1}, \frac{k \cdot (-4) + 1 \cdot 4}{k + 1}, \frac{k \cdot (-5) + 1 \cdot 5}{k + 1} \right) \]
This simplifies to:
\[ P = \left( \frac{3k + 2}{k + 1}, \frac{-4k + 4}{k + 1}, \frac{-5k + 5}{k + 1} \right) \]

### Step 4: Set the z-coordinate to zero
Since point \( P \) lies on the XY-plane, we set the z-coordinate equal to zero:
\[ \frac{-5k + 5}{k + 1} = 0 \]

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

### Step 6: Determine the ratio
The ratio in which the line segment is divided is \( k:1 = 1:1 \).

### Final Answer
The line segment joining the points \( (2, 4, 5) \) and \( (3, -4, -5) \) is divided by the XY-plane in the ratio \( 1:1 \).

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