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

Options

A. 3 : 4 internally
B. 4 : 5 externally
C. 2 : 6 internally
D. 6 : 7 externally

Answer

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

### Step 1: Identify the Coordinates
Let \( A = (x_1, y_1, z_1) = (2, 4, -3) \) and \( B = (x_2, y_2, z_2) = (-3, 5, 4) \).

### Step 2: Understand the Condition
The line segment is divided by the XY-plane, which means the z-coordinate of the point of division (let's call it point \( R \)) will be zero. Therefore, we need to find the ratio \( m:n \) such that the z-coordinate of \( R \) is zero.

### Step 3: Use the Section Formula
The coordinates of point \( R \) dividing the line segment \( AB \) in the ratio \( m:n \) can be calculated using the section formula:
- The z-coordinate of \( R \) is given by:
\[ z = \frac{mz_2 + nz_1}{m+n} \]
Substituting the coordinates:
\[ z = \frac{m \cdot 4 + n \cdot (-3)}{m+n} \]

### Step 4: Set the z-coordinate to Zero
Since we want the z-coordinate to be zero, we set up the equation:
\[ 0 = \frac{m \cdot 4 + n \cdot (-3)}{m+n} \]
This implies:
\[ m \cdot 4 - n \cdot 3 = 0 \]

### Step 5: Solve for the Ratio
Rearranging the equation gives:
\[ 4m = 3n \]
Dividing both sides by \( n \) gives:
\[ \frac{m}{n} = \frac{3}{4} \]
Thus, the ratio \( m:n \) is \( 3:4 \).

### Step 6: Conclusion
The ratio in which the line segment joining the points \( (2, 4, -3) \) and \( (-3, 5, 4) \) is divided by the XY-plane is \( 3:4 \).

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