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 = ______

# 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 = ______

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### Solution:

Since $\mathrm{\ell }=\mathrm{m}=\mathrm{n}=\frac{1}{\sqrt{3}}$

$\therefore$ Equation of line are $\frac{\mathrm{x}-2}{1/\sqrt{3}}=\frac{\mathrm{y}+1}{1/\sqrt{3}}=\frac{\mathrm{z}-2}{1/\sqrt{3}}$

$\therefore$Any point on the line is

$\because$ Q lies on the plane 2x + y + z = 9

$\begin{array}{l}⇒2\left(\mathrm{r}+2\right)+\left(\mathrm{r}-1\right)+\left(\mathrm{r}+2\right)=9\\ ⇒4\mathrm{r}+5=9⇒\mathrm{r}=1\\ \therefore \mathrm{PQ}=\sqrt{\left(3-2{\right)}^{2}+\left(0+1{\right)}^{2}+\left(3-2{\right)}^{2}}=\sqrt{3}\end{array}$  +91

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