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 PQ equals units.

Step-by-step solution

1. Direction Cosines: If a line makes equal angles with coordinate axes, its direction ratios are proportional to (1,1,1) . 2. Equation of the Line: Parametric form is: x-2 1 = y-1 1 = z-2 1 =t or, x=2+t, y=1+t, z=2+t 3. Point of Intersection with Plane: Substituting x=2+t, y=1+t, z=2+t into 2 x+y+z=9 : gathered 2(2+t)+(1+t)+(2+t)=9 4+2 t+1+t+2+t=9 4 t+7=9 t= 2 4 = 1 2 gathered Substituting t= 1 2 back, the coordinates of Q are: Q= (2+ 1 2 , 1+ 1 2 , 2+ 1 2 )= ( 5 2 , 3 2 , 5 2 ) 4. Length of PQ: Using the distance