Question

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

Options

•  a : b

•  b : c

• c : a

• c : b

Answer 1

 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.

\[\therefore \frac{- b\lambda + c}{\lambda + 1} = 0\]
\[ \Rightarrow - b\lambda + c = 0\]
\[ \Rightarrow \lambda = \frac{c}{b}\]

Hence, the xz-plane divides AB in the ratio c : b