Given that the points `A(2,3,4),B(-1,2,-3)` and `C(-4,1,-10)` are collinear. Then the ratio in which C divides AB is

Options

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

Answer

To find the ratio in which point C divides the line segment AB, we can follow these steps:

### Step 1: Identify the coordinates of points A, B, and C
- A(2, 3, 4)
- B(-1, 2, -3)
- C(-4, 1, -10)

### Step 2: Assume the ratio in which C divides AB
Let the ratio in which C divides AB be \( \lambda : 1 \).

### Step 3: Use the section formula
According to the section formula, the coordinates of point C can be expressed in terms of the coordinates of points A and B as follows:
- The x-coordinate of C:
\[ x_C = \frac{\lambda x_B + x_A}{\lambda + 1} \]
- The y-coordinate of C:
\[ y_C = \frac{\lambda y_B + y_A}{\lambda + 1} \]
- The z-coordinate of C:
\[ z_C = \frac{\lambda z_B + z_A}{\lambda + 1} \]

### Step 4: Substitute the coordinates into the equations
Substituting the coordinates of points A, B, and C into the equations:
1. For the x-coordinate:
\[ -4 = \frac{\lambda (-1) + 2}{\lambda + 1} \]
Simplifying this gives:
\[ -4(\lambda + 1) = -\lambda + 2 \]
\[ -4\lambda - 4 = -\lambda + 2 \]
\[ -4\lambda + \lambda = 2 + 4 \]
\[ -3\lambda = 6 \implies \lambda = -2 \]

2. For the y-coordinate:
\[ 1 = \frac{\lambda (2) + 3}{\lambda + 1} \]
Simplifying this gives:
\[ 1(\lambda + 1) = 2\lambda + 3 \]
\[ \lambda + 1 = 2\lambda + 3 \]
\[ 1 - 3 = 2\lambda - \lambda \]
\[ -2 = \lambda \implies \lambda = -2 \]

3. For the z-coordinate:
\[ -10 = \frac{\lambda (-3) + 4}{\lambda + 1} \]
Simplifying this gives:
\[ -10(\lambda + 1) = -3\lambda + 4 \]
\[ -10\lambda - 10 = -3\lambda + 4 \]
\[ -10\lambda + 3\lambda = 4 + 10 \]
\[ -7\lambda = 14 \implies \lambda = -2 \]

### Step 5: Conclusion
The value of \( \lambda \) is consistently found to be -2. This indicates that point C divides the line segment AB externally in the ratio of 2:1.

### Final Answer
C divides AB in the ratio of 2:1 externally.