Correct Answer - Option 2 : 1 ∶ 2

**Concept:**

**Section Formula:** Section formula is used to determine the coordinate of a point that divides a line into two parts such that ratio of their length is m ∶ n

1. Let P and Q be the given two points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) respectively and M(x, y, z) be the point dividing the line segment PQ internally in the ratio m: n

**2. Internal Section Formula:** When the line segment is divided internally in the ration m: n, we use this formula.\(\rm (x, y, z)=(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}, \frac{mz_2+nz_1}{m+n})\)

**Calculation:**

Here, the point R(5, 4, -6) divides the points P(3, 2, -4) and Q(9, 8, -10)

Let, required ratio be k:1, Then the coordinates of R are

(\(\rm \frac{9k+3}{k+1}, \frac{8k+2}{k+1},\frac{-10k-4}{k+1}\))

But coordinates of R are (5, 4, -6)

∴\(\rm \frac{9k+3}{k+1}=5\)

\(\rm ⇒ 9k+3=5k+5\)

\(\rm ⇒ 4k=2\)

⇒ k = 1/2

∴ k ∶ 1 = \(\frac 1 2 :1\)

= 1 ∶ 2

Hence, option (2) is correct.