Question

Find an equation for the line that passes through the point P(2,3,1) and is paralllel to the line of intersection of the planes x+2y-3z=4 and x-2y+z=0.

Answer 1

Let the diection ratios of the required line be proportional to a, b, c. As it lies on both the planes. So, their normals are perpndicular to it.

Therefore,

$a+2b-3c=0$

$a-2b+c=0$

Using cross multiplication, we get,

$\frac{a}{1}=\frac{b}{1}=\frac{c}{1}$

Thus, required line passes through the point P(2,3,1) and its direction ratios are proportional to 1,1,1. Hence, its equation is

$\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-1}{1}$