Question

Find the coordinates of the point where the line through $(5,1,6)$ and $(3,4,1)$ crosses the $ZX$-plane

Answer 1

The line passing through the points $(5,1,6)$ and $(3,4,1)$ is given by,
$\frac{x-5}{3-5}=\frac{y-1}{4-1}=\frac{z-6}{1-6}$
$\Rightarrow$ $\frac{x-5}{-2}=\frac{y-1}{3}=\frac{z-6}{-5}=k$ (say)
$\Rightarrow$ $x=5-2k,y=3k+1,z=6-5k$
Any point on the line is of the form $P(5-2k,3k+1,6-5k)$.
Since the line passes through $ZX$-plane,
$\Rightarrow Y-$coordinate of point $P$ will be $0$ $\Rightarrow 3k+1=0\Rightarrow k=-\frac{1}{3}$
Therefore, the required point is $(\frac{17}{3},0,\frac{23}{3})$.