Question

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

Answer 1

We know that the line passing through two points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ is

$\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}$

So, the equation of line passing through (5,1,6) and (3,4,1) is

$\frac{x-5}{3-5}=\frac{y-1}{4-1}=\frac{z-6}{1-6}$

$\Rightarrow \frac{x-5}{-2}=\frac{y-1}{3}=\frac{z-6}{-5}=k\left(say\right)$

Any point on this line is
$x=-2k+5,y=3k+1,z=-5k+6$

The line crosses ZX -plane when y = 0,

$\Rightarrow 3k+1=0$

$\Rightarrow k=-\frac{1}{3}$

So, the required point is

$x=-2(-\frac{1}{3})+5$,

$y=3(-\frac{1}{3})+1$,

$z=-5(-\frac{1}{3})+6$

$\Rightarrow x=\frac{17}{3},y=0,z=\frac{23}{3}$

Hence, the required coordinates of the point are : $(\frac{17}{3},0,\frac{23}{3})$