What is the ratio in which the line joining the points (2,4,5) and (3,5,-4) is internally divided by the xy-plane?

Options

A. `5:4`
B. `3:4`
C. `1:2`
D. `7:5`

Explanation

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

### Step 1: Understand the problem
The xy-plane is defined by the equation \( z = 0 \). We need to find the point on the line segment joining points \( A \) and \( B \) where the z-coordinate is 0.

### Step 2: Use the section formula
Let the ratio in which the line segment is divided be \( k:1 \). According to the section formula, the coordinates of the point \( P \) that divides the line segment joining points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) in the ratio \( m:n \) are given by:

\[ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]

### Step 3: Assign coordinates
For our points:
- \( A(2, 4, 5) \) implies \( x_1 = 2, y_1 = 4, z_1 = 5 \)
- \( B(3, 5, -4) \) implies \( x_2 = 3, y_2 = 5, z_2 = -4 \)

### Step 4: Set up the equation for the z-coordinate
We need the z-coordinate of point \( P \) to be 0:

\[ z = \frac{k(-4) + 1(5)}{k + 1} = 0 \]

### Step 5: Solve the equation
Setting the equation equal to zero:

\[ 0 = \frac{-4k + 5}{k + 1} \]

Cross-multiplying gives:

\[ 0 = -4k + 5 \]

Rearranging this gives:

\[ 4k = 5 \implies k = \frac{5}{4} \]

### Step 6: Determine the ratio
The ratio \( k:1 \) is \( \frac{5}{4}:1 \), which simplifies to \( 5:4 \).

### Final Answer
Thus, the ratio in which the line joining the points \( (2, 4, 5) \) and \( (3, 5, -4) \) is internally divided by the xy-plane is \( \mathbf{5:4} \).