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 = ______

Solution:

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

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

$\Rightarrow x - 2 = y + 1 = z - 2 = r (say)$

$∴$ Any point on the line is $Q \equiv (r + 2, r - 1, r + 2)$

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

$\begin{array}{l} \Rightarrow 2 (r + 2) + (r - 1) + (r + 2) = 9 \\ \Rightarrow 4 r + 5 = 9 \Rightarrow r = 1 \\ ∴ PQ = \sqrt{(3 - 2)^{2} + (0 + 1)^{2} + (3 - 2)^{2}} = \sqrt{3} \end{array}$

Solution:

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