Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

For the line to lie in the plane, it must be parallel to the plane and any point on the line must satisfy the equation of the plane.

The direction ratios of the line are $1,1,2$ and the direction ratios of the normal to the plane are $2,-4,1$.

Checking the condition for parallelism:
$1(2)+1(-4)+2(1)=2-4+2=0$
Thus, the line is parallel to the plane.

A point on the given line is $(4,2,-\mathrm{m})$. Since the line lies in the plane, this point must satisfy the equation of the plane $2x-4y+z=7$.

Substituting the coordinates of the point into the plane's equation:
$2(4)-4(2)+(-\mathrm{m})=7$

$8-8-\mathrm{m}=7$

$-\mathrm{m}=7$

$\mathrm{m}=-7$

Answer: -7