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:

  1. 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 $

$\Rightarrow \quad 4 r+5=9 \Rightarrow r=1$

$\Rightarrow \quad Q(3,0,3)$

$\therefore \quad P Q=\sqrt{(3-2)^{2}+(0+1)^{2}+(3-2)^{2}}=\sqrt{3}$



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