If `A(3, 2, -4), B(5, 4, -6) and C(9, 8, -10)` are three collinear points, then the ratio in which point C divides AB.

Answer

To find the ratio in which point C divides the line segment AB, we can use the section formula in three-dimensional geometry. The section formula states that if a point C divides the line segment joining points A and B in the ratio \( m:n \), then the coordinates of C can be expressed as:

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

Where \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) are the coordinates of points A and B respectively.

### Step 1: Identify the coordinates of points A, B, and C
Given:
- \( A(3, 2, -4) \)
- \( B(5, 4, -6) \)
- \( C(9, 8, -10) \)

### Step 2: Set up the equations using the section formula
Assume that point C divides AB in the ratio \( \lambda:1 \). Therefore, we can express the coordinates of C as:

\[ C_x = \frac{5\lambda + 3}{\lambda + 1}, \quad C_y = \frac{4\lambda + 2}{\lambda + 1}, \quad C_z = \frac{-6\lambda - 4}{\lambda + 1} \]

### Step 3: Equate the coordinates of C
Now, we equate the coordinates of C to the given coordinates of point C:

1. For the x-coordinate:
\[ \frac{5\lambda + 3}{\lambda + 1} = 9 \]

2. For the y-coordinate:
\[ \frac{4\lambda + 2}{\lambda + 1} = 8 \]

3. For the z-coordinate:
\[ \frac{-6\lambda - 4}{\lambda + 1} = -10 \]

### Step 4: Solve the equations
Let's solve the first equation:

\[ 5\lambda + 3 = 9(\lambda + 1) \]
\[ 5\lambda + 3 = 9\lambda + 9 \]
\[ 5\lambda - 9\lambda = 9 - 3 \]
\[ -4\lambda = 6 \implies \lambda = -\frac{3}{2} \]

### Step 5: Interpret the result
The negative value of \( \lambda \) indicates that point C divides the segment AB externally. The ratio in which C divides AB can be expressed as:

\[ \text{Ratio} = \frac{|\lambda|}{1} = \frac{3/2}{1} = 3:2 \]

### Final Answer
Thus, point C divides the line segment AB in the ratio of \( 3:2 \) externally.

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