A point R with x-coordinate 4 lies on the line segment joining the points P(2, – 3, 4) and Q(8, 0, 10). Find the coordinates of the point R

# A point R with x-coordinate 4 lies on the line segment joining the points P(2, - 3, 4) and Q(8, 0, 10). Find the coordinates of the point R

1. A

(4, -2, -6)

2. B

(4, 2,6)

3. C

(4, -2, 6)

4. D

None of these

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### Solution:

Let the point R (x,y, z) divides PQ in the ratio k:1.

$\begin{array}{l}⇒\frac{\mathrm{k}×8+1×2}{\mathrm{k}+1}=4\\ ⇒\frac{8\mathrm{k}+2}{\mathrm{k}+1}=4\\ ⇒8\mathrm{k}+2=4\mathrm{k}+4\\ ⇒8\mathrm{k}-4\mathrm{k}=4-2\\ ⇒4\mathrm{k}=2\\ ⇒\mathrm{k}=\frac{1}{2}\\ ⇒\mathrm{k}:1=1:2\end{array}$

Hence, the point R divides PQ internally in the ratio 1 : 2.

Therefore, y-coordinate of $\mathrm{R}=\left(\frac{1×0+2×\left(-3\right)}{1+2}\right)=\frac{-6}{3}=-2$

and z-coordinate of $\mathrm{R}=\left(\frac{1×10+2×4}{1+2}\right)=\frac{10+8}{3}=\frac{18}{3}=6$

Hence, coordinate of R are (4, -2, 6).

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