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.
$$\begin{gathered} \therefore \frac{10\lambda +4}{\lambda +1}=8 \\ \Rightarrow 10\lambda +4=8\lambda +8 \\ \Rightarrow 2\lambda =4 \\ \therefore \lambda =2 \end{gathered}$$
Hence, the coordinates of C are as follows:
$$\begin{gathered} \left(\frac{8\lambda +2}{\lambda +1},\frac{-3}{\lambda +1},\frac{10\lambda +4}{\lambda +1}\right) \\ \Rightarrow \left(\frac{8\times 2+2}{2+1},\frac{-3}{2+1},\frac{10\times 2+4}{2+1}\right) \\ \Rightarrow \left(\frac{18}{3},\frac{-3}{3},\frac{24}{3}\right) \\ \therefore \left(6,-1,8\right) \end{gathered}$$
Hence, the coordinates of C are (6, −1, 8).