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Question
The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)
Options
internally in the ratio 2 : 3
externally in the ratio 2 : 3
internally in the ratio 3 : 2
externally in the ratio 3 : 2
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Solution
`\text{ externally in the ratio 2: 3 } `
\[\text{ Let the XY - plane divide the line segment joining points }P\left( - 1, 3, 4 \right) \text{ and } Q\left( 2, - 5, 6 \right) \text{ in the ratio k: 1 }. \]
\[\text { Using the section formula, the coordinates of the point of intersection are given by } \]
\[\left( \frac{k\left( 2 \right) - 1}{k + 1}, \frac{k\left( - 5 \right) + 3}{k + 1}, \frac{k\left( 6 \right) + 4}{k + 1} \right)\]
\[\text { On the XY - plane, the Z - coordinate of any point is zero } . \]
\[ \Rightarrow \frac{k\left( 6 \right) + 4}{k + 1} = 0\]
\[ \Rightarrow 6k + 4 = 0\]
\[ \Rightarrow k = - \frac{2}{3}\]
\[\text{ Thus, the XY - plane divides the line segment joining the given points in the ratio 2: 3 externally } . \]
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