The co-ordinates of the point where the line through \(\mathrm{A}(3,4,1)\) and \(\mathrm{B}(5,1,6)\) crosses the \(x y\)-plane are

Let \(\mathrm{A}\left(x_1, y_1, \mathrm{z}_1\right)=\mathrm{A}(3,4,1)\)
\(\mathrm{B}\left(x_2, y_2, z_2\right)=\mathrm{B}(5,1,6)\)
The equation of line passing through the points \(\left(x_1, y_1, z_1\right)\) and \(\left(x_2, y_2, z_2\right)\) is given by
\(\begin{array}{ll}
& \frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1} \\
\therefore & \frac{x-3}{5-3}=\frac{y-4}{1-4}=\frac{z-1}{6-1} \\
\therefore & \frac{x-3}{2}=\frac{y-4}{-3}=\frac{z-1}{5}
\end{array}\)
Since the line crosses the XY plane, \(\mathrm{z}=0\)
\(\therefore \quad \frac{x-3}{2}=\frac{y-4}{-3}=\frac{-1}{5}\)
\(\therefore \quad \frac{x-3}{2}=\frac{-1}{5}\) and \(\frac{y-4}{-3}=\frac{-1}{5}\)
\(\Rightarrow x=\frac{13}{5}\) and \(y=\frac{23}{5}\)
Required point is \(\left(\frac{13}{5}, \frac{23}{5}, 0\right)\)