If the points `A(3,2,-4),\ B(9,8,-10)a n d\ C(5,4,-6)` are collinear, find the ratio in which C divides AB.

Options

A. `1 : 3` internally
B. `1 : 3` externally
C. `3 : 1` internally
D. `3 : 1 ` externally

Answer

To find the ratio in which point C divides the line segment AB, we can use the section formula. The section formula states that if a point C divides the line segment joining points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) in the ratio m:n, then the coordinates of point C can be expressed as:

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

Given the points:
- A(3, 2, -4)
- B(9, 8, -10)
- C(5, 4, -6)

Let the ratio in which C divides AB be m:n.

### Step 1: Set up the equations using the x-coordinates
Using the x-coordinates, we have:
\[ 5 = \frac{9m + 3n}{m+n} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 5(m + n) = 9m + 3n \]
Expanding this, we get:
\[ 5m + 5n = 9m + 3n \]

### Step 3: Rearrange the equation
Rearranging the equation to isolate terms involving m and n:
\[ 5m + 5n - 9m - 3n = 0 \]
This simplifies to:
\[ -4m + 2n = 0 \]

### Step 4: Solve for the ratio
Rearranging gives:
\[ 4m = 2n \quad \Rightarrow \quad \frac{m}{n} = \frac{2}{4} = \frac{1}{2} \]

### Step 5: Conclusion
Thus, the ratio in which C divides AB is 1:2.