#### Question

The *xy*-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)

internally in the ratio 2 : 3

externally in the ratio 2 : 3

internally in the ratio 3 : 2

externally in the ratio 3 : 2

#### Solution

`\text{ externally in the ratio 2: 3 } `

\[\text{ Let the XY - plane divide the line segment joining points }P\left( - 1, 3, 4 \right) \text{ and } Q\left( 2, - 5, 6 \right) \text{ in the ratio k: 1 }. \]

\[\text { Using the section formula, the coordinates of the point of intersection are given by } \]

\[\left( \frac{k\left( 2 \right) - 1}{k + 1}, \frac{k\left( - 5 \right) + 3}{k + 1}, \frac{k\left( 6 \right) + 4}{k + 1} \right)\]

\[\text { On the XY - plane, the Z - coordinate of any point is zero } . \]

\[ \Rightarrow \frac{k\left( 6 \right) + 4}{k + 1} = 0\]

\[ \Rightarrow 6k + 4 = 0\]

\[ \Rightarrow k = - \frac{2}{3}\]

\[\text{ Thus, the XY - plane divides the line segment joining the given points in the ratio 2: 3 externally } . \]