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If A=[(x,y,z)],B=[(a,h,g),(h,b,f),(g,f,c...

If `A=[(x,y,z)],B=[(a,h,g),(h,b,f),(g,f,c)]` and `C=[(x),(y),(z)]`
Then `ABC=O` if

A

`[ax^(2)+by^(2)+cz^(2)+2gxy+2fyz+2czx]=O`

B

`[ax^(2)+cy^(2)+bz^(2)+xy+yz+zx]=O`

C

`[ax^(2)+by^(2)+cz^(2)+2hxy+2by+2cz]=O`

D

`[ax^(2)+by^(2)+cz^(2)+2gzx+2hxy+2fyz]=O`

Text Solution

Verified by Experts

The correct Answer is:
D
`AB==[(x, y, z)][(a,h,g),(h,b,f),(g,f,c)]`
`=[xa+yh+zg, xh+yb+zf, xg+yf+zc)]`
Now `ABC=O`
`=[xa+yh+zg, xh+yb+zf, xg+yf+zc][(x),(y),(z)]=O`
`implies[ax^(2)+hxy+gxz+hxy+y^(2)b+fzy+gxz+yfz+z^(2)c]=O`
`implies[ax^(2)+by^(2)+cz^(2)=2gzx+2hxy+2fyz]=O`
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