Paragraph: Let a , b and c be three real numbers satisfying [ array lll a & b & c array ] [ array lll 1 & 9 & 7 8 & 2 & 7 7 & 3 & 7 array ]= [ array lll 0 & 0 & 0 array ] Question: If the point P(a, b, c) , with reference to Eq. (E), lies on the plane 2 x+y+z=1 , then the value of 7 a+b+c is

Step-by-step solution

aligned & Given, [a b c]_ 1 3 [ array lll 1 & 9 & 7 8 & 2 & 7 7 & 3 & 7 array ]_ 3 3 = [ array lll 0 & 0 & 0 array ] & [ array c a+8 b+7 c 9 a+2 b+3 c 7 a+7 b+7 c array ]= [ array l 0 0 0 array ] & a+8 b+7 c=0 & 9 a+2 b+3 c=0 & a+b+c=0 & aligned On multiplying Eq. (iii) by 2 , then subtract from Eq. (ii), we get 7 a+c=0 Again multiplying Eq. (iii) by 3 , then subtract from Eq. (ii), we get array ll & 6 a-b=0 & b=6 a and c=-7 a array As (a, b, c) lies on 2 x+y+z=1 array lc & 2 a+b+c=1 & 2 a+6 a-7 a=1 & a=1, b=6 and