Let L_1 be the line of intersection of the planes given by the equation- 2 x+3 y+z=4 and x+2 y+z=5 . Let L_2 be the line passing through the point P(2,-1,3) and parallel to L_1. Let M denote the plane given by the equation- 2 x+y-2 z=6 Suppose that the line L_2 meets the plane M at the point Q. Let R be the foot of the perpendicular drawn from P to the plane M. Then which of the following statements is (are) TRUE?
C. The area of P Q R is 3 2 234
Let L_1: r = a +t b b = | array ccc i & j & k \\ 2 & 3 & 1 \\ 1 & 2 & 1 array |= i (1)- j (1)+ k (1) Dr's of L_1: aligned & L_1: x+1 1 = y-0 -1 = z-6 1 \\ & \& L_2: x-2 1 = y+1 -1 = z-3 1 = \\ & M: 2 x+y-2 z-6=0 aligned Let point on L _2( +2,- -1, +3) aligned & 2( +2)- -1-2 -6-6=0 \\ & 2 +4-3 -13=0 \\ & =-9 \\ & Q(-7,8,-6) aligned Line PR: x-2 2 = y+1 1 = z-3 -2 = R (2 +2, -1,-2 +3) Put in plane aligned & 2(2 +2)+ -1-2(-2 +3)-6=0 \\ & 4 +4+ -1+4 -6-6=0 \\ & 9 -9=0 =1 \\ & R (4,0,1) \\ & P (2,-1,3) \& Q (-7,8,-6) \\ & PQ = 81+81+81 =9 3 \\ & Q (-7,8,-6) \& R (4,0,1) aligned QR = 121+64+49 = 234 Area ( PQR ) aligned & = 1 2 | QP QR | \\ & = 1 2 | array ccc i & j & k \\ 9 & -9 & 9 \\ 11 & -8 & 7 array | \\ & = 3 2 234 \\ & PQ =-9 i +9 j -9 k =-9( i - j + k ) \\ & PR =2 i + j -2 k aligned aligned & = | PQ PR PQ PR | \\ & = 9 9 3 3 = 1 3 3 \\ & = ^ -1 ( 1 3 3 ) aligned
Related: Mathematics — Three Dimensional Geometry · All PYQ Banks