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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
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Solution
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
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