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

• A. $1$
• B. $\sqrt 2$
• C. $\sqrt 3$
• D. $2$

Answer 1

Answer: (C)

$\sqrt 3$

Answer 2

Since, \hspace15mm l = m = n \frac{1}{\sqrt 3}
$\therefore$ Equations of line are $\frac{x-2}{1/ \sqrt 3 } = \frac{y+1}{1/ \sqrt 3 } = \frac{z-2}{1/ \sqrt 3 }$
$\Rightarrow$ \hspace40mm x - 2 = y +1 = z - 2 = r \, \, \, [say]
$\therefore$ Any point on the line is
\hspace30mm Q = (r+2,r-1,r+2)
$\because Q$ lies on the plane $2x+ y +z = 9$
$\therefore$ \hspace5mm 2(r+2)+(r-1)+(r+2) = 9
$\Rightarrow$ \hspace40mm 4r + 5 = 9
$\Rightarrow$ \hspace50mm r = 1
$\Rightarrow$ \hspace15mm Q(3,\, 0,\, 3)
$\therefore$ $PQ= {\sqrt{(3-2)^2+(0+1)^2+(3-2)^2}} = \sqrt 3$