A plane is parallel to two lines, whose direction ratios are 1,0,-1 and -1,1,0 and it contains the point (1,1,1) . If it cuts co-ordinate axes (X, Y, Z - axes resp.) at A , B , C , then the volume of the tetrahedron OABC is cu. units.

Step-by-step solution

Equation of the plane passing through (1,1,1) is given as a (x-1)+ b (y-1)+ c ( z -1)=0 As the plane is parallel to the lines having direction ratios 1,0,-1 and -1,1,0 , we get aligned & a-c=0 and -a+b=0 & a=b=c aligned From (i) and (ii), we get aligned & x-1+y-1+z-1=0 & x+y+z=3 x 3 + y 3 + z 3 =1 aligned Co-ordinates of A, B, C are (3,0,0),(0,3,0) and (0,0,3) respectively. aligned Volume of tetrahedron OABC & = 1 6 | array lll 3 & 0 & 0 0 & 3 & 0 0 & 0 & 3 array | & = 1 6 27 & = 9 2 cu. units aligned