A line with positive direction cosines passes through the point P… - Vidyadip

Question

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:

$1$
$\sqrt{2}$
$\sqrt{3}$
$2$

✓

Answer

$\sqrt{3}$

Solution:
D.C of the line are $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$
Any point on the line at a distance tt from P(2, -1, 2) is
$\Big(2+\frac{\text{t}}{\sqrt{3}},-1+\frac{\text{t}}{\sqrt{3}},2+\frac{\text{t}}{\sqrt{3}}\Big)$
which lies on $2\text{x} + \text{y + z} = 9$
$\Rightarrow\text{t}=\sqrt{3}$

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