A straight line drawn from the point P(1,3,2), parallel to the line x-2 1 = y-4 2 = z-6 1 , intersects the plane L_1: x-y+3 z=6 at the point Q. Another straight line which passes through Q and is perpendicular to the plane L_1 intersects the plane L_2: 2 x-y+z=-4 at the point R. Then which of the following statements is (are) TRUE?
C. The centroid of the triangle P Q R is ( 4 3 , 14 3 , 5 3 )
aligned & line : x-1 1 = y-3 2 = z-2 1 \\ & (x, y, z)=( +1,2 +3, +2) aligned Put in L_1: x-y+3 z=6 aligned & ( +1)-(2 +3)+3( +2)=6 \\ & 2 =2 =1 \\ & Q=(2,5,3) aligned line : x-2 1 = y-5 -1 = z-3 3 (x, y, z)=(t+2,5-t, 3 t+3) Put in L_2: 2 x-y+z=-4 aligned & 2(t+4)-(5-t)+(3 t+3)=-4 \\ & 6 t=-6 t=-1 \\ & R=(1,6,0) aligned aligned & Perimeter = 6 + 13 + 11 \\ & Centroid = ( 4 3 , 14 3 , 5 3 ) aligned
Related: Mathematics — Three Dimensional Geometry · All PYQ Banks