Question

XOZ-plane divides the join of (2, 3, 1) and (6, 7, 1) in the ratio
(a) 3 : 7
(b) 2 : 7
(c) –3 : 7
(d) –2 : 7

Answer 1

(c) −3:7

Let A$\equiv$(2, 3, 1) and B$\equiv$(6, 7, 1)
Let the line joining A and B be divided by the xz-plane at point P in the ratio $\lambda :1$.
Then, we have,

P$\equiv \left(\frac{6\lambda +2}{\lambda +1},\frac{7\lambda +3}{\lambda +1},\frac{\lambda +1}{\lambda +1}\right)$

Since P lies on the xz-plane, the y-coordinate of P will be zero.

$$\begin{gathered} \therefore \frac{7\lambda +3}{\lambda +1}=0 \\ \Rightarrow 7\lambda +3=0 \\ \Rightarrow \lambda =\frac{-3}{7} \end{gathered}$$
Hence, the xz-plane divides AB in the ratio $-$3 : 7