A line with positive direction cosines passes through the point P(2,-1,2) and makes equal …

Question

A line with positive direction cosines passes through the point $P(2,-1,2)$ and makes equal angles with the coordinate axes. If the line meets the plane $2x+y+z=9$ at point $Q$, then the length $PQ$ equals

Accepted Answer

Equal direction cosines $\Rightarrow$ equal direction ratios: point on line $(h,h-3,h)$ (since $y=h-3$ keeps ratios equal to the $x,z$ increments from $P$). Substituting into the plane: $2h+(h-3)+h=9\Rightarrow4h=12\Rightarrow h=3$, giving $Q=(3,0,3)$.
$$PQ=\sqrt{(3-2)^2+(0+1)^2+(3-2)^2}=\sqrt3.$$

Solutions · 1