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+6y +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

The correct option is C

$\sqrt{3}$

Since, $l=m=n=\frac{1}{\sqrt{3}}$

Equations of line are$\frac{x-2}{\frac{1}{\sqrt{3}}}=\frac{y+1}{\frac{1}{\sqrt{3}}}=\frac{z-2}{\frac{1}{\sqrt{3}}}\Rightarrow x-2=y+1=z-2=r$
$\therefore$ Any point on the line is
$Q\equiv (r+2,r-1,r+2)\therefore$Q lies on the plane $2x+y+z=9\therefore 2(r+2)+(r-1)+(r+2)=9\Rightarrow 4r+5=9\Rightarrow r=1\Rightarrow Q(3,0,3)\therefore PQ=\sqrt{(3-2{)}^2+(0+1{)}^2+(3-2{)}^2}=\sqrt{3}$