The distance of the origin from the point of intersection of the line $\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and the plane $2x+y-z=2$ is.

Line : $\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}=\lambda ;\text{ Plane }:2x+y-z=2$

$\therefore$ point on line $(2\lambda ,3\lambda +2,4\lambda +3)$

$\therefore 2\left(2\lambda \right)+(3\lambda +2)-(4\lambda +3)=2\Rightarrow \lambda =1$

$\therefore$ distance from origin $=\sqrt{2^2+5^2+7^2}=\sqrt{78}$