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Question
Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, –8) is divided by the yz-plane.
Solution
Let A = (4, 8, 10) , B = (6, 10,\[-\]8)
Let the line joining A and B be divided by the YZ-plane at point P in the ratio \[\lambda: 1\]
∴ P=\[\left( \frac{6\lambda + 4}{\lambda + 1}, \frac{10\lambda + 8}{\lambda + 1}, \frac{- 8\lambda + 10}{\lambda + 1} \right)\]
Since P lies on the YZ-plane, the x-coordinate of P will be zero.
\[\therefore \frac{6\lambda + 4}{\lambda + 1} = 0\]
\[ \Rightarrow 6\lambda + 4 = 0\]
\[ \Rightarrow 6\lambda = - 4\]
\[ \therefore \lambda = \frac{- 2}{3}\]
Hence, the YZ-plane divides AB in the ratio 2:3 (externally)
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