Question

The ratio in which the line joining the points (a, b, c) and (–a, –c, –b) is divided by the xy-plane is
(a) a : b
(b) b : c
(c) c : a
(d) c : b

Answer 1

(d) c : b

Let A$\equiv$(a, b, c) and B$\equiv$($-$a, $-$c, $-$b)
Let the line joining A and B be divided by the xy-plane at point P in the ratio $\lambda :1$.
Then, we have,

P$\equiv \left(\frac{-a\lambda +a}{\lambda +1},\frac{-c\lambda +b}{\lambda +1},\frac{-b\lambda +c}{\lambda +1}\right)$

Since P lies on the xy-plane, the z-coordinate of P will be zero.

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