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

We know that the equation of the line that passes through the points $(x_{1},y_{1},z_{1})$and $(x_{2},y_{2},z_{2})$ is given by the relation;

$\frac{x-x_{1}}{x_{2}-x_{1}} = \frac{y-y_{1}}{y_{2}-y_{1}} = \frac{z-z_{1}}{z_{2}-z_{1}}$

and the line passing through the points, $\frac{x-5}{3-5} =\frac{y-1}{4-1} = \frac{z-6}{1-6}$

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

$\implies x=5-2k,\ y=3k+1,\ z=6-5k$

And any point on the line is of the form $(5-2k,3k+1,6-5k)$.

So, the equation of the YZ plane is $x=0$

Since the line passes through YZ- plane,

we have then,

$5-2k = 0$

$\Rightarrow k =\frac{5}{2}$

or  $3k+1 = 3(\frac{5}{2})+1 = \frac{17}{2}$   and  $6-5k= 6-5(\frac{5}{2}) = \frac{-13}{2}$

So, therefore the required point is $\left ( 0,\frac{17}{2},\frac{-13}{2} \right )$

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