A point R with x-coordinate 4 lies on the line segment joining th… - Vidyadip

Question

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. [Hint Suppose R divides PQ in the ratio k : 1. The coordinates of the point R are given by $\Big(\frac{8\text{k}+2}{\text{k}+1},\frac{-3}{\text{k}+1},\frac{10\text{k}+4}{\text{k}+1}\Big)]$

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Answer

The coordinates of points P and Q are given as P(2, –3, 4) and Q(8, 0, 10). Let R divide line segment PQ in the ratio k : 1. Hence, by section formula, the coordinates of point R are given by $\Big(\frac{\text{k}(8)+2}{\text{k}+1},\frac{\text{k}(0)-3}{\text{k}+1},\frac{\text{k}(10)+4}{\text{k}+1}\Big)=\Big(\frac{8\text{k}+2}{\text{k}+1},\frac{-3}{\text{k}+1},\frac{10\text{k}+4}{\text{k}+1}\Big)$ It is given that the x-coordinate of point R is 4. $\therefore\frac{8\text{k}+2}{\text{k}+1}=4$ $\Rightarrow8\text{k}+2=4\text{k}+4$ $\Rightarrow4\text{k}=2$ $\Rightarrow\text{k}=\frac{1}{2}$ Therefore, the coordinates of point R are $\begin{pmatrix}4,\frac{-3}{\frac{1}{2}+1},\frac{10\big(\frac{1}{2}\big)+4}{\frac{1}{2}+1}\end{pmatrix}=(4,-2,6)$

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