First slide

Introduction to 3-D Geometry

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

Easy

A

(4, -2, -6)
B

(4, 2,6)
C

(4, -2, 6)
D

None of these

Solution

Let the point R (x,y, z) divides PQ in the ratio k:1.
$\begin{array}{l} \Rightarrow \frac{k \times 8 + 1 \times 2}{k + 1} = 4 \\ \Rightarrow \frac{8 k + 2}{k + 1} = 4 \\ \Rightarrow 8 k + 2 = 4 k + 4 \\ \Rightarrow 8 k - 4 k = 4 - 2 \\ \Rightarrow 4 k = 2 \\ \Rightarrow k = \frac{1}{2} \\ \Rightarrow k : 1 = 1 : 2 \end{array}$
Hence, the point R divides PQ internally in the ratio 1 : 2.
Therefore, y-coordinate of $R = (\frac{1 \times 0 + 2 \times (- 3)}{1 + 2}) = \frac{- 6}{3} = - 2$
and z-coordinate of $R = (\frac{1 \times 10 + 2 \times 4}{1 + 2}) = \frac{10 + 8}{3} = \frac{18}{3} = 6$
Hence, coordinate of R are (4, -2, 6).

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