Question

A point C with z-coordinate 8 lies on the line segment joining the points A(2, –3, 4) and B(8, 0, 10). Find its coordinates.

Answer 1

Suppose C divides AB in the ratio \[\lambda: 1\]Then, the coordinates of C are\[\left( \frac{8\lambda + 2}{\lambda + 1}, \frac{- 3}{\lambda + 1}, \frac{10\lambda + 4}{\lambda + 1} \right)\] The z-coordinate of C is 8.

\[\therefore \frac{10\lambda + 4}{\lambda + 1} = 8\]

\[ \Rightarrow 10\lambda + 4 = 8\lambda + 8\]

\[ \Rightarrow 2\lambda = 4\]

\[ \therefore \lambda = 2\] 

Hence, the coordinates of C are as follows: 

\[\left( \frac{8\lambda + 2}{\lambda + 1}, \frac{- 3}{\lambda + 1}, \frac{10\lambda + 4}{\lambda + 1} \right)\]
\[ \Rightarrow \left( \frac{8 \times 2 + 2}{2 + 1}, \frac{- 3}{2 + 1}, \frac{10 \times 2 + 4}{2 + 1} \right)\]
\[ \Rightarrow \left( \frac{18}{3}, \frac{- 3}{3}, \frac{24}{3} \right)\]
\[ \therefore \left( 6, - 1, 8 \right)\]

Hence, the coordinates of C are (6, −1, 8).