Let R be such that the lines L _1: x+11 1 = y +21 2 = z +29 3 and L_2: x+16 3 = y+11 2 = z+4 intersect. Let R _1 be the point of intersection of L _1 and L _2. Let O =(0,0,0), and n denote a unit normal vector to the plane containing both the lines L _1 and L _2. Match each entry in List-I to the correct entry in List-II. The correct option is
C. ( P ) (3) ( Q ) (4) ( R ) (1) ( S ) (5)
aligned & L _1: x +11 1 = y +21 2 = z +29 3 = a \\ & L _2: x +16 3 = y +11 2 = z +4 = b aligned aligned & x=a-11=3 b-16 a-3 b=-5...(1) \\ & y=2 a-21=2 b-11 2 a-2 b=10...(2) \\ & z=3 a-29=b r-4 3 a-b =25...(3) aligned from (1) & (2) a =10, ~b =5 Now from (3) 3(10)-5 =25 =1 aligned & R _1 (-1,-1,1) \\ & OR _1=- i - j + k \\ & n = | array lrr i & j & k \\ 1 & 2 & 3 \\ 3 & 2 & 1 array |=-4 i -(-8) j -4 k \\ & n =-4 i +8 j +4 k =-4( i -2 j + k ) \\ & n = 4( i -2 j + k ) 4 6 = ( i -2 j + k ) 6 \\ & OR _1 n = (- i - j + k ) ( i -2 j + k 6 )= 2 6 = 4 6 = 2 3 aligned
Related: Mathematics — Three Dimensional Geometry · All PYQ Banks