# Find the ratio in which the join of the points P(3, 2, -4) and Q(9,8, -10) is divided by the point R(5, 4, -6)

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Find the ratio in which the join of the points P(3, 2, -4) and Q(9,8, -10) is divided by the point R(5, 4, -6)
1. 2 ∶ 1
2. 1 ∶ 2
3. 3 ∶ 1
4. 2 ∶ 3
5. None of these

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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 (x1, y1, z1) and (x2, y2, z2) 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.