The coordinates of the point where the line through the points A (3, 4, 1) and B (5, 1, 6) crosses the XY - plane are

Equation of the line through the given points is

$\begin{array}{r}\frac{x-3}{5-3}=\frac{y-4}{1-4}=\frac{z-1}{6-1}\\ \Rightarrow \frac{x-3}{2}=\frac{y-4}{-3}=\frac{z-1}{5}\end{array}$

Any point on this line can be taken as

$(3+2\lambda ,4-3\lambda ,1+5\lambda )$

If this point lies on XY-plane then the z-coordinate is zero

$\Rightarrow 1+5\lambda =0\Rightarrow \lambda =-\frac{1}{5}$

Thus the required coordinates of the point are

$\left(3-\frac{2}{5},4-3\left(-\frac{1}{5}\right),0\right)\equiv \left(\frac{13}{5},\frac{23}{5},0\right)$