#### Question

Ratio in which the *xy*-plane divides the join of (1, 2, 3) and (4, 2, 1) is

3 : 1 internally

3 : 1 externally

1 : 2 internally

2 : 1 externally

#### Solution

` 3: 1 \text{ externally } `

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

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

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

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

\[ \Rightarrow \frac{k\left( 1 \right) + 3}{k + 1} = 0\]

\[ \Rightarrow k + 3 = 0\]

\[ \Rightarrow k = - 3 = - \frac{3}{1}\]

\[\text{ Thus, the XY - plane divides the line segment joining the given points in the ratio 3: 1 externally } . \]