The shortest distance between the z-axis and the line x+y+2z-3=0=2x+3y+4z-4, is

Question

The shortest distance between the $z$-axis and the line $x+y+2z-3=0=2x+3y+4z-4$, is

Accepted Answer

Family through the line: $(x+y+2z-3)+\lambda(2x+3y+4z-4)=0$. Parallel to the $z$-axis (zero $z$-coefficient): $2+4\lambda=0\Rightarrow\lambda=-\tfrac12$.
Plane: $y+2=0$ (i.e. $y=-2$).
$$\text{Distance from the }z\text{-axis (points }(0,0,t)\text{)}=\frac{|0+2|}{1}=2.$$

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