3D Geometry 3 Question 45
45. 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 $2 x+y+z=9$ at point $Q$. The length of the line segment $P Q$ equals
(2009)
(a) 1
(b) $\sqrt{2}$
(c) $\sqrt{3}$
(d) 2
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Answer:
Correct Answer: 45. (c)
Solution:
- Since,
$ l=m=n=\frac{1}{\sqrt{3}} $
$\therefore$ Equations of line are $\frac{x-2}{1 / \sqrt{3}}=\frac{y+1}{1 / \sqrt{3}}=\frac{z-2}{1 / \sqrt{3}}$
$\Rightarrow$ $ \quad x-2=y+1=z-2=r $
[say]
$\therefore$ Any point on the line is
$ Q \equiv(r+2, r-1, r+2) $
$\because Q$ lies on the plane $2 x+y+z=9$
$ \therefore \quad 2(r+2)+(r-1)+(r+2)=9 $
$\Rightarrow \quad 4 r+5=9 \Rightarrow r=1$
$\Rightarrow \quad Q(3,0,3)$
$\therefore \quad P Q=\sqrt{(3-2)^{2}+(0+1)^{2}+(3-2)^{2}}=\sqrt{3}$
