In which ratio the line segment joining the points
(3, 4, 10) and (-3, 2, 5) is divided by x-axis?

Options

A. `2 :1` internally
B. `2 : 1` externally
C. `1 : 2` internally
D. `1 : 2` externally

Answer

To find the ratio in which the line segment joining the points \( P(3, 4, 10) \) and \( Q(-3, 2, 5) \) is divided by the x-axis, we can follow these steps:

### Step 1: Identify the coordinates of the points
We have two points:
- \( P(3, 4, 10) \)
- \( Q(-3, 2, 5) \)

### Step 2: Determine the equation of the x-axis
The x-axis is defined by the conditions:
- \( y = 0 \)
- \( z = 0 \)

### Step 3: Set up the section formula
Let the point \( R(x, 0, 0) \) be the point where the line segment \( PQ \) intersects the x-axis. According to the section formula, if a point divides the line segment joining two points \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) in the ratio \( m:n \), then the coordinates of the point \( R \) can be given by:

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

### Step 4: Substitute the coordinates into the section formula
We want \( R \) to lie on the x-axis, hence \( y = 0 \) and \( z = 0 \).

1. **For the y-coordinate**:
\[ \frac{m \cdot 2 + n \cdot 4}{m+n} = 0 \]
This gives us:
\[ m \cdot 2 + n \cdot 4 = 0 \implies 2m + 4n = 0 \implies m = -2n \]

2. **For the z-coordinate**:
\[ \frac{m \cdot 5 + n \cdot 10}{m+n} = 0 \]
This gives us:
\[ m \cdot 5 + n \cdot 10 = 0 \implies 5m + 10n = 0 \implies m = -2n \]

### Step 5: Find the ratio
From both conditions, we have \( m = -2n \). Thus, the ratio \( m:n \) can be expressed as:
\[ m:n = -2n:n = -2:1 \]
Since the ratio must be positive, we can express it in positive terms as:
\[ 2:1 \]

### Conclusion
The line segment joining the points \( P(3, 4, 10) \) and \( Q(-3, 2, 5) \) is divided by the x-axis in the ratio \( 2:1 \).