Question

Find the ratio in which the line joining (2, 4, 5) and (3, 5, 4) is divided by the yz-plane.

Answer 1

Let A$\equiv$(2, 4, 5) and B$\equiv$(3, 5, 4)
Let the line joining A and B be divided by the yz-plane at point P in the ratio $\lambda :1$.
Then, we have:
P$\equiv \left(\frac{3\lambda +2}{\lambda +1},\frac{5\lambda +4}{\lambda +1},\frac{4\lambda +5}{\lambda +1}\right)$
Since P lies on the yz-plane, the x-coordinate of P will be zero.
$$\begin{gathered} \therefore \frac{3\lambda +2}{\lambda +1}=0 \\ \Rightarrow 3\lambda +2=0 \\ \therefore \lambda =\frac{-2}{3} \end{gathered}$$
Hence, the yz-plane divides AB in the ratio 2:3 (externally).