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 \( A(3, 4, 10) \) and \( B(-3, 2, 5) \) is divided by the x-axis, we can follow these steps:

### Step 1: Understand the Points and the Division
The points given are:
- \( A(3, 4, 10) \)
- \( B(-3, 2, 5) \)

The x-axis is represented by the equation \( y = 0 \) and \( z = 0 \). Therefore, any point on the x-axis can be represented as \( (x, 0, 0) \).

### Step 2: Assume the Ratio
Let the x-axis divide the line segment \( AB \) in the ratio \( k:1 \). This means we can express the coordinates of the point \( P \) where the line segment intersects the x-axis as:

\[ P\left( \frac{-3k + 3}{k + 1}, \frac{2k + 4}{k + 1}, \frac{5k + 10}{k + 1} \right) \]

### Step 3: Set the y and z Coordinates to Zero
Since point \( P \) lies on the x-axis, the y and z coordinates must be equal to zero:
1. For the y-coordinate:
\[ \frac{2k + 4}{k + 1} = 0 \]
This implies:
\[ 2k + 4 = 0 \implies 2k = -4 \implies k = -2 \]

2. For the z-coordinate:
\[ \frac{5k + 10}{k + 1} = 0 \]
This will also lead to the same value of \( k \) since:
\[ 5k + 10 = 0 \implies 5k = -10 \implies k = -2 \]

### Step 4: Determine the Ratio
Since \( k = -2 \), the ratio in which the x-axis divides the segment \( AB \) is:
\[ k : 1 = -2 : 1 \]
This indicates that the division is external.

### Final Answer
Thus, the line segment joining the points \( (3, 4, 10) \) and \( (-3, 2, 5) \) is divided by the x-axis in the ratio \( 2:1 \) externally.

---