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

MCQ

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 :

A
$1$
B
$\sqrt{2}$
✓
$\sqrt{3}$
D
$2$

✓

Answer

Correct option: C.

$\sqrt{3}$

$D.C$ 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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