The $$xy$$ -plane divides the line joining the points $$\left( { - 1,\,3,\,4} \right)\left( {2,\, - 5,\,6} \right)$$

Solution :
Suppose $$xy$$ -plane divides the line joining the given points in the ratio $$\lambda :1.$$
The coordinate of the point of division are
$$\left( {\frac{{2\lambda - 1}}{{\lambda + 1}},\,\frac{{ - 5\lambda + 3}}{{\lambda + 1}},\,\frac{{6\lambda + 4}}{{\lambda + 1}}} \right)$$
This point lies on $$xy$$ -plane
$$\frac{{6\lambda + 4}}{{\lambda + 1}} = 0 \Rightarrow \lambda = - \frac{3}{2}$$
Hence, $$xy$$ -plane divides externally in the ratio $$3 : 2.$$