Given that `P (3,2,-4), Q(5,4,-6), R (9,8-10)` are collinear, find the ratio in which Q divides PR.

Answer

To find the ratio in which point Q divides the line segment PR, we can use the section formula. The coordinates of the points are given as follows:

- P(3, 2, -4)
- Q(5, 4, -6)
- R(9, 8, -10)

Let Q divide PR in the ratio m:n. According to the section formula, the coordinates of point Q can be expressed as:

\[ Q_x = \frac{m \cdot R_x + n \cdot P_x}{m + n} \]
\[ Q_y = \frac{m \cdot R_y + n \cdot P_y}{m + n} \]
\[ Q_z = \frac{m \cdot R_z + n \cdot P_z}{m + n} \]

Where \(Q_x, Q_y, Q_z\) are the coordinates of Q, and \(P_x, P_y, P_z\) and \(R_x, R_y, R_z\) are the coordinates of P and R respectively.

### Step 1: Set up the equations for each coordinate
Using the coordinates provided:
1. For the x-coordinate:
\[ 5 = \frac{m \cdot 9 + n \cdot 3}{m + n} \]
2. For the y-coordinate:
\[ 4 = \frac{m \cdot 8 + n \cdot 2}{m + n} \]
3. For the z-coordinate:
\[ -6 = \frac{m \cdot (-10) + n \cdot (-4)}{m + n} \]

### Step 2: Solve the x-coordinate equation
Multiply both sides of the x-coordinate equation by \(m + n\):
\[ 5(m + n) = 9m + 3n \]
Expanding gives:
\[ 5m + 5n = 9m + 3n \]
Rearranging terms:
\[ 5m - 9m = 3n - 5n \]
This simplifies to:
\[ -4m = -2n \quad \Rightarrow \quad 4m = 2n \quad \Rightarrow \quad \frac{m}{n} = \frac{2}{4} = \frac{1}{2} \]

### Step 3: Conclusion
Thus, the ratio in which Q divides PR is:
\[ m:n = 1:2 \]