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`

Answer

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: Identify the coordinates of the points
The coordinates of the points are:
- Point A: \( (x_1, y_1, z_1) = (2, 4, 5) \)
- Point B: \( (x_2, y_2, z_2) = (3, 5, -4) \)

### Step 2: Understand the xy-plane condition
The xy-plane is defined by the equation \( z = 0 \). Therefore, we need to find the point on the line segment \( AB \) where \( z = 0 \).

### Step 3: 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 \( AB \) in the ratio \( k:1 \) are given by:
\[ P\left( \frac{kx_2 + x_1}{k + 1}, \frac{ky_2 + y_1}{k + 1}, \frac{kz_2 + z_1}{k + 1} \right) \]

### Step 4: Set up the equation for z-coordinate
For the point \( P \) to lie on the xy-plane, we set the z-coordinate to 0:
\[ \frac{kz_2 + z_1}{k + 1} = 0 \]
Substituting \( z_1 = 5 \) and \( z_2 = -4 \):
\[ \frac{k(-4) + 5}{k + 1} = 0 \]

### Step 5: Solve for k
Setting the numerator equal to zero:
\[ k(-4) + 5 = 0 \]
\[ -4k + 5 = 0 \]
\[ 4k = 5 \]
\[ k = \frac{5}{4} \]

### Step 6: Determine the ratio
The ratio in which the line segment is divided is \( k:1 = \frac{5}{4}:1 \). To express this as a ratio of integers, we can multiply both sides by 4:
\[ 5:4 \]

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

---