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

$\text{Given: line through } P(2,-1,2),\ \text{direction cosines all positive}$ $\text{equal angles with the axes};\quad \text{plane } 2x + y + z = 9,\ \text{meet at } Q$ $\text{To find: the length } PQ$ $1.\ \text{direction from equal direction cosines}$ $2.\ \text{general point on the line}$ $3.\ \text{meet the plane} \Rightarrow Q \Rightarrow |PQ|$ $\text{plane } 2x + y + z = 9, \quad \vec{n} = (2,1,1)$ $l = m = n, \quad l^2 + m^2 + n^2 = 1$ $3l^2 = 1 \Rightarrow l = m = n = \dfrac{1}{\sqrt{3}}$ $\text{direction ratios} = (1,\ 1,\ 1)$ $\dfrac{x-2}{1} = \dfrac{y+1}{1} = \dfrac{z-2}{1} = t$ $\text{general point } (2+t,\ -1+t,\ 2+t)$ $2(2+t) + (-1+t) + (2+t) = 9$ $5 + 4t = 9$ $4t = 4 \Rightarrow t = 1$ $Q = (2+1,\ -1+1,\ 2+1) = (3,\ 0,\ 3)$ $2(3) + 0 + 3 = 9 \ \Rightarrow \ \text{verified}$ $\vec{PQ} = Q - P = (1,\ 1,\ 1)$ $|PQ| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$ $\boxed{\text{Answer: (3)}\ \ PQ = \sqrt{3}}$